:: ENUMSET1 semantic presentation
Lemma1:
for b1, b2, b3 being set holds
( b1 in union {b2,{b3}} iff ( b1 in b2 or b1 = b3 ) )
definition
let c
1, c
2, c
3 be
set ;
func {c1,c2,c3} -> set means :
Def1:
:: ENUMSET1:def 1
for b
1 being
set holds
( b
1 in a
4 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines { ENUMSET1:def 1 :
for b
1, b
2, b
3, b
4 being
set holds
( b
4 = {b1,b2,b3} iff for b
5 being
set holds
( b
5 in b
4 iff not ( not b
5 = b
1 & not b
5 = b
2 & not b
5 = b
3 ) ) );
definition
let c
1, c
2, c
3, c
4 be
set ;
func {c1,c2,c3,c4} -> set means :
Def2:
:: ENUMSET1:def 2
for b
1 being
set holds
( b
1 in a
5 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines { ENUMSET1:def 2 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
( b
5 = {b1,b2,b3,b4} iff for b
6 being
set holds
( b
6 in b
5 iff not ( not b
6 = b
1 & not b
6 = b
2 & not b
6 = b
3 & not b
6 = b
4 ) ) );
definition
let c
1, c
2, c
3, c
4, c
5 be
set ;
func {c1,c2,c3,c4,c5} -> set means :
Def3:
:: ENUMSET1:def 3
for b
1 being
set holds
( b
1 in a
6 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 & not b
1 = a
5 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 & not b2 = c5 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines { ENUMSET1:def 3 :
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
6 = {b1,b2,b3,b4,b5} iff for b
7 being
set holds
( b
7 in b
6 iff not ( not b
7 = b
1 & not b
7 = b
2 & not b
7 = b
3 & not b
7 = b
4 & not b
7 = b
5 ) ) );
definition
let c
1, c
2, c
3, c
4, c
5, c
6 be
set ;
func {c1,c2,c3,c4,c5,c6} -> set means :
Def4:
:: ENUMSET1:def 4
for b
1 being
set holds
( b
1 in a
7 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 & not b
1 = a
5 & not b
1 = a
6 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 & not b2 = c5 & not b2 = c6 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines { ENUMSET1:def 4 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
7 = {b1,b2,b3,b4,b5,b6} iff for b
8 being
set holds
( b
8 in b
7 iff not ( not b
8 = b
1 & not b
8 = b
2 & not b
8 = b
3 & not b
8 = b
4 & not b
8 = b
5 & not b
8 = b
6 ) ) );
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func {c1,c2,c3,c4,c5,c6,c7} -> set means :
Def5:
:: ENUMSET1:def 5
for b
1 being
set holds
( b
1 in a
8 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 & not b
1 = a
5 & not b
1 = a
6 & not b
1 = a
7 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 & not b2 = c5 & not b2 = c6 & not b2 = c7 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines { ENUMSET1:def 5 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( b
8 = {b1,b2,b3,b4,b5,b6,b7} iff for b
9 being
set holds
( b
9 in b
8 iff not ( not b
9 = b
1 & not b
9 = b
2 & not b
9 = b
3 & not b
9 = b
4 & not b
9 = b
5 & not b
9 = b
6 & not b
9 = b
7 ) ) );
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
set ;
func {c1,c2,c3,c4,c5,c6,c7,c8} -> set means :
Def6:
:: ENUMSET1:def 6
for b
1 being
set holds
( b
1 in a
9 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 & not b
1 = a
5 & not b
1 = a
6 & not b
1 = a
7 & not b
1 = a
8 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 & not b2 = c5 & not b2 = c6 & not b2 = c7 & not b2 = c8 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 & not b3 = c8 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 & not b3 = c8 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines { ENUMSET1:def 6 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
9 = {b1,b2,b3,b4,b5,b6,b7,b8} iff for b
10 being
set holds
( b
10 in b
9 iff not ( not b
10 = b
1 & not b
10 = b
2 & not b
10 = b
3 & not b
10 = b
4 & not b
10 = b
5 & not b
10 = b
6 & not b
10 = b
7 & not b
10 = b
8 ) ) );
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8, c
9 be
set ;
func {c1,c2,c3,c4,c5,c6,c7,c8,c9} -> set means :
Def7:
:: ENUMSET1:def 7
for b
1 being
set holds
( b
1 in a
10 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 & not b
1 = a
5 & not b
1 = a
6 & not b
1 = a
7 & not b
1 = a
8 & not b
1 = a
9 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 & not b2 = c5 & not b2 = c6 & not b2 = c7 & not b2 = c8 & not b2 = c9 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 & not b3 = c8 & not b3 = c9 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 & not b3 = c8 & not b3 = c9 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines { ENUMSET1:def 7 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set holds
( b
10 = {b1,b2,b3,b4,b5,b6,b7,b8,b9} iff for b
11 being
set holds
( b
11 in b
10 iff not ( not b
11 = b
1 & not b
11 = b
2 & not b
11 = b
3 & not b
11 = b
4 & not b
11 = b
5 & not b
11 = b
6 & not b
11 = b
7 & not b
11 = b
8 & not b
11 = b
9 ) ) );
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8, c
9, c
10 be
set ;
func {c1,c2,c3,c4,c5,c6,c7,c8,c9,c10} -> set means :
Def8:
:: ENUMSET1:def 8
for b
1 being
set holds
( b
1 in a
11 iff not ( not b
1 = a
1 & not b
1 = a
2 & not b
1 = a
3 & not b
1 = a
4 & not b
1 = a
5 & not b
1 = a
6 & not b
1 = a
7 & not b
1 = a
8 & not b
1 = a
9 & not b
1 = a
10 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff not ( not b2 = c1 & not b2 = c2 & not b2 = c3 & not b2 = c4 & not b2 = c5 & not b2 = c6 & not b2 = c7 & not b2 = c8 & not b2 = c9 & not b2 = c10 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 & not b3 = c8 & not b3 = c9 & not b3 = c10 ) ) ) & ( for b3 being set holds
( b3 in b2 iff not ( not b3 = c1 & not b3 = c2 & not b3 = c3 & not b3 = c4 & not b3 = c5 & not b3 = c6 & not b3 = c7 & not b3 = c8 & not b3 = c9 & not b3 = c10 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines { ENUMSET1:def 8 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
set holds
( b
11 = {b1,b2,b3,b4,b5,b6,b7,b8,b9,b10} iff for b
12 being
set holds
( b
12 in b
11 iff not ( not b
12 = b
1 & not b
12 = b
2 & not b
12 = b
3 & not b
12 = b
4 & not b
12 = b
5 & not b
12 = b
6 & not b
12 = b
7 & not b
12 = b
8 & not b
12 = b
9 & not b
12 = b
10 ) ) );
theorem Th1: :: ENUMSET1:1
canceled;
theorem Th2: :: ENUMSET1:2
canceled;
theorem Th3: :: ENUMSET1:3
canceled;
theorem Th4: :: ENUMSET1:4
canceled;
theorem Th5: :: ENUMSET1:5
canceled;
theorem Th6: :: ENUMSET1:6
canceled;
theorem Th7: :: ENUMSET1:7
canceled;
theorem Th8: :: ENUMSET1:8
canceled;
theorem Th9: :: ENUMSET1:9
canceled;
theorem Th10: :: ENUMSET1:10
canceled;
theorem Th11: :: ENUMSET1:11
canceled;
theorem Th12: :: ENUMSET1:12
canceled;
theorem Th13: :: ENUMSET1:13
canceled;
theorem Th14: :: ENUMSET1:14
canceled;
theorem Th15: :: ENUMSET1:15
canceled;
theorem Th16: :: ENUMSET1:16
canceled;
theorem Th17: :: ENUMSET1:17
canceled;
theorem Th18: :: ENUMSET1:18
canceled;
theorem Th19: :: ENUMSET1:19
canceled;
theorem Th20: :: ENUMSET1:20
canceled;
theorem Th21: :: ENUMSET1:21
canceled;
theorem Th22: :: ENUMSET1:22
canceled;
theorem Th23: :: ENUMSET1:23
canceled;
theorem Th24: :: ENUMSET1:24
canceled;
theorem Th25: :: ENUMSET1:25
canceled;
theorem Th26: :: ENUMSET1:26
canceled;
theorem Th27: :: ENUMSET1:27
canceled;
theorem Th28: :: ENUMSET1:28
canceled;
theorem Th29: :: ENUMSET1:29
canceled;
theorem Th30: :: ENUMSET1:30
canceled;
theorem Th31: :: ENUMSET1:31
canceled;
theorem Th32: :: ENUMSET1:32
canceled;
theorem Th33: :: ENUMSET1:33
canceled;
theorem Th34: :: ENUMSET1:34
canceled;
theorem Th35: :: ENUMSET1:35
canceled;
theorem Th36: :: ENUMSET1:36
canceled;
theorem Th37: :: ENUMSET1:37
canceled;
theorem Th38: :: ENUMSET1:38
canceled;
theorem Th39: :: ENUMSET1:39
canceled;
theorem Th40: :: ENUMSET1:40
canceled;
theorem Th41: :: ENUMSET1:41
theorem Th42: :: ENUMSET1:42
theorem Th43: :: ENUMSET1:43
Lemma13:
for b1, b2, b3, b4 being set holds {b1,b2,b3,b4} = {b1,b2} \/ {b3,b4}
theorem Th44: :: ENUMSET1:44
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b1} \/ {b2,b3,b4}
theorem Th45: :: ENUMSET1:45
theorem Th46: :: ENUMSET1:46
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b1,b2,b3} \/ {b4}
Lemma17:
for b1, b2, b3, b4, b5 being set holds {b1,b2,b3,b4,b5} = {b1,b2,b3} \/ {b4,b5}
theorem Th47: :: ENUMSET1:47
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b2,b3,b4,b5} = {b1} \/ {b2,b3,b4,b5}
theorem Th48: :: ENUMSET1:48
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b2,b3,b4,b5} = {b1,b2} \/ {b3,b4,b5}
theorem Th49: :: ENUMSET1:49
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b2,b3,b4,b5} = {b1,b2,b3} \/ {b4,b5} by Lemma17;
theorem Th50: :: ENUMSET1:50
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b2,b3,b4,b5} = {b1,b2,b3,b4} \/ {b5}
Lemma22:
for b1, b2, b3, b4, b5, b6 being set holds {b1,b2,b3,b4,b5,b6} = {b1,b2,b3} \/ {b4,b5,b6}
theorem Th51: :: ENUMSET1:51
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b2,b3,b4,b5,b6} = {b1} \/ {b2,b3,b4,b5,b6}
theorem Th52: :: ENUMSET1:52
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2} \/ {b3,b4,b5,b6}
theorem Th53: :: ENUMSET1:53
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2,b3} \/ {b4,b5,b6} by Lemma22;
theorem Th54: :: ENUMSET1:54
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4} \/ {b5,b6}
theorem Th55: :: ENUMSET1:55
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4,b5} \/ {b6}
Lemma26:
for b1, b2, b3, b4, b5, b6, b7 being set holds {b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4} \/ {b5,b6,b7}
theorem Th56: :: ENUMSET1:56
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1} \/ {b2,b3,b4,b5,b6,b7}
theorem Th57: :: ENUMSET1:57
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2} \/ {b3,b4,b5,b6,b7}
theorem Th58: :: ENUMSET1:58
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3} \/ {b4,b5,b6,b7}
theorem Th59: :: ENUMSET1:59
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4} \/ {b5,b6,b7} by Lemma26;
theorem Th60: :: ENUMSET1:60
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4,b5} \/ {b6,b7}
theorem Th61: :: ENUMSET1:61
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4,b5,b6} \/ {b7}
Lemma31:
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds {b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8}
theorem Th62: :: ENUMSET1:62
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1} \/ {b2,b3,b4,b5,b6,b7,b8}
theorem Th63: :: ENUMSET1:63
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2} \/ {b3,b4,b5,b6,b7,b8}
theorem Th64: :: ENUMSET1:64
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3} \/ {b4,b5,b6,b7,b8}
theorem Th65: :: ENUMSET1:65
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8} by Lemma31;
theorem Th66: :: ENUMSET1:66
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4,b5} \/ {b6,b7,b8}
theorem Th67: :: ENUMSET1:67
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4,b5,b6} \/ {b7,b8}
theorem Th68: :: ENUMSET1:68
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8} = {b1,b2,b3,b4,b5,b6,b7} \/ {b8}
theorem Th69: :: ENUMSET1:69
theorem Th70: :: ENUMSET1:70
for b
1, b
2 being
set holds
{b1,b1,b2} = {b1,b2}
theorem Th71: :: ENUMSET1:71
for b
1, b
2, b
3 being
set holds
{b1,b1,b2,b3} = {b1,b2,b3}
theorem Th72: :: ENUMSET1:72
for b
1, b
2, b
3, b
4 being
set holds
{b1,b1,b2,b3,b4} = {b1,b2,b3,b4}
theorem Th73: :: ENUMSET1:73
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b1,b2,b3,b4,b5} = {b1,b2,b3,b4,b5}
theorem Th74: :: ENUMSET1:74
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4,b5,b6}
theorem Th75: :: ENUMSET1:75
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
{b1,b1,b2,b3,b4,b5,b6,b7} = {b1,b2,b3,b4,b5,b6,b7}
theorem Th76: :: ENUMSET1:76
for b
1 being
set holds
{b1,b1,b1} = {b1}
theorem Th77: :: ENUMSET1:77
for b
1, b
2 being
set holds
{b1,b1,b1,b2} = {b1,b2}
theorem Th78: :: ENUMSET1:78
for b
1, b
2, b
3 being
set holds
{b1,b1,b1,b2,b3} = {b1,b2,b3}
theorem Th79: :: ENUMSET1:79
for b
1, b
2, b
3, b
4 being
set holds
{b1,b1,b1,b2,b3,b4} = {b1,b2,b3,b4}
theorem Th80: :: ENUMSET1:80
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b1,b1,b2,b3,b4,b5} = {b1,b2,b3,b4,b5}
theorem Th81: :: ENUMSET1:81
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
{b1,b1,b1,b2,b3,b4,b5,b6} = {b1,b2,b3,b4,b5,b6}
theorem Th82: :: ENUMSET1:82
for b
1 being
set holds
{b1,b1,b1,b1} = {b1}
theorem Th83: :: ENUMSET1:83
for b
1, b
2 being
set holds
{b1,b1,b1,b1,b2} = {b1,b2}
theorem Th84: :: ENUMSET1:84
for b
1, b
2, b
3 being
set holds
{b1,b1,b1,b1,b2,b3} = {b1,b2,b3}
theorem Th85: :: ENUMSET1:85
for b
1, b
2, b
3, b
4 being
set holds
{b1,b1,b1,b1,b2,b3,b4} = {b1,b2,b3,b4}
theorem Th86: :: ENUMSET1:86
for b
1, b
2, b
3, b
4, b
5 being
set holds
{b1,b1,b1,b1,b2,b3,b4,b5} = {b1,b2,b3,b4,b5}
theorem Th87: :: ENUMSET1:87
for b
1 being
set holds
{b1,b1,b1,b1,b1} = {b1}
theorem Th88: :: ENUMSET1:88
for b
1, b
2 being
set holds
{b1,b1,b1,b1,b1,b2} = {b1,b2}
theorem Th89: :: ENUMSET1:89
for b
1, b
2, b
3 being
set holds
{b1,b1,b1,b1,b1,b2,b3} = {b1,b2,b3}
theorem Th90: :: ENUMSET1:90
for b
1, b
2, b
3, b
4 being
set holds
{b1,b1,b1,b1,b1,b2,b3,b4} = {b1,b2,b3,b4}
theorem Th91: :: ENUMSET1:91
for b
1 being
set holds
{b1,b1,b1,b1,b1,b1} = {b1}
theorem Th92: :: ENUMSET1:92
for b
1, b
2 being
set holds
{b1,b1,b1,b1,b1,b1,b2} = {b1,b2}
theorem Th93: :: ENUMSET1:93
for b
1, b
2, b
3 being
set holds
{b1,b1,b1,b1,b1,b1,b2,b3} = {b1,b2,b3}
theorem Th94: :: ENUMSET1:94
for b
1 being
set holds
{b1,b1,b1,b1,b1,b1,b1} = {b1}
theorem Th95: :: ENUMSET1:95
for b
1, b
2 being
set holds
{b1,b1,b1,b1,b1,b1,b1,b2} = {b1,b2}
theorem Th96: :: ENUMSET1:96
for b
1 being
set holds
{b1,b1,b1,b1,b1,b1,b1,b1} = {b1}
theorem Th97: :: ENUMSET1:97
canceled;
theorem Th98: :: ENUMSET1:98
for b
1, b
2, b
3 being
set holds
{b1,b2,b3} = {b1,b3,b2}
theorem Th99: :: ENUMSET1:99
for b
1, b
2, b
3 being
set holds
{b1,b2,b3} = {b2,b1,b3}
theorem Th100: :: ENUMSET1:100
for b
1, b
2, b
3 being
set holds
{b1,b2,b3} = {b2,b3,b1}
theorem Th101: :: ENUMSET1:101
canceled;
theorem Th102: :: ENUMSET1:102
for b
1, b
2, b
3 being
set holds
{b1,b2,b3} = {b3,b2,b1}
theorem Th103: :: ENUMSET1:103
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b1,b2,b4,b3}
theorem Th104: :: ENUMSET1:104
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b1,b3,b2,b4}
theorem Th105: :: ENUMSET1:105
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b1,b3,b4,b2}
theorem Th106: :: ENUMSET1:106
canceled;
theorem Th107: :: ENUMSET1:107
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b1,b4,b3,b2}
theorem Th108: :: ENUMSET1:108
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b2,b1,b3,b4}
Lemma65:
for b1, b2, b3, b4 being set holds {b1,b2,b3,b4} = {b2,b3,b1,b4}
theorem Th109: :: ENUMSET1:109
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b2,b1,b4,b3}
theorem Th110: :: ENUMSET1:110
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b2,b3,b1,b4} by Lemma65;
theorem Th111: :: ENUMSET1:111
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b2,b3,b4,b1}
theorem Th112: :: ENUMSET1:112
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b2,b4,b1,b3}
theorem Th113: :: ENUMSET1:113
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b2,b4,b3,b1}
Lemma66:
for b1, b2, b3, b4 being set holds {b1,b2,b3,b4} = {b3,b2,b1,b4}
theorem Th114: :: ENUMSET1:114
canceled;
theorem Th115: :: ENUMSET1:115
canceled;
theorem Th116: :: ENUMSET1:116
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b3,b2,b1,b4} by Lemma66;
theorem Th117: :: ENUMSET1:117
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b3,b2,b4,b1}
theorem Th118: :: ENUMSET1:118
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b3,b4,b1,b2}
theorem Th119: :: ENUMSET1:119
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b3,b4,b2,b1}
theorem Th120: :: ENUMSET1:120
canceled;
theorem Th121: :: ENUMSET1:121
canceled;
theorem Th122: :: ENUMSET1:122
canceled;
theorem Th123: :: ENUMSET1:123
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b4,b2,b3,b1}
theorem Th124: :: ENUMSET1:124
canceled;
theorem Th125: :: ENUMSET1:125
for b
1, b
2, b
3, b
4 being
set holds
{b1,b2,b3,b4} = {b4,b3,b2,b1}
Lemma68:
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds {b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8,b9}
theorem Th126: :: ENUMSET1:126
canceled;
theorem Th127: :: ENUMSET1:127
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1} \/ {b2,b3,b4,b5,b6,b7,b8,b9}
theorem Th128: :: ENUMSET1:128
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2} \/ {b3,b4,b5,b6,b7,b8,b9}
theorem Th129: :: ENUMSET1:129
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3} \/ {b4,b5,b6,b7,b8,b9}
theorem Th130: :: ENUMSET1:130
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8,b9} by Lemma68;
theorem Th131: :: ENUMSET1:131
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5} \/ {b6,b7,b8,b9}
theorem Th132: :: ENUMSET1:132
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5,b6} \/ {b7,b8,b9}
theorem Th133: :: ENUMSET1:133
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5,b6,b7} \/ {b8,b9}
theorem Th134: :: ENUMSET1:134
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4,b5,b6,b7,b8} \/ {b9}
theorem Th135: :: ENUMSET1:135
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set holds
{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10} = {b1,b2,b3,b4,b5,b6,b7,b8,b9} \/ {b10}
theorem Th136: :: ENUMSET1:136
for b
1, b
2, b
3 being
set holds
( b
1 <> b
2 & b
1 <> b
3 implies
{b1,b2,b3} \ {b1} = {b2,b3} )