:: SETFAM_1 semantic presentation
definition
let c
1 be
set ;
func meet c
1 -> set means :
Def1:
:: SETFAM_1:def 1
for b
1 being
set holds
( b
1 in a
2 iff for b
2 being
set holds
( b
2 in a
1 implies b
1 in b
2 ) )
if a
1 <> {} otherwise a
2 = {} ;
existence
( not ( c1 <> {} & ( for b1 being set holds
not for b2 being set holds
( b2 in b1 iff for b3 being set holds
( b3 in c1 implies b2 in b3 ) ) ) ) & not ( not c1 <> {} & ( for b1 being set holds
not b1 = {} ) ) )
uniqueness
for b1, b2 being set holds
( ( c1 <> {} & ( for b3 being set holds
( b3 in b1 iff for b4 being set holds
( b4 in c1 implies b3 in b4 ) ) ) & ( for b3 being set holds
( b3 in b2 iff for b4 being set holds
( b4 in c1 implies b3 in b4 ) ) ) implies b1 = b2 ) & ( not c1 <> {} & b1 = {} & b2 = {} implies b1 = b2 ) )
consistency
for b1 being set holds
verum
;
end;
:: deftheorem Def1 defines meet SETFAM_1:def 1 :
for b
1 being
set for b
2 being
set holds
( ( b
1 <> {} implies ( b
2 = meet b
1 iff for b
3 being
set holds
( b
3 in b
2 iff for b
4 being
set holds
( b
4 in b
1 implies b
3 in b
4 ) ) ) ) & ( not b
1 <> {} implies ( b
2 = meet b
1 iff b
2 = {} ) ) );
theorem Th1: :: SETFAM_1:1
canceled;
theorem Th2: :: SETFAM_1:2
theorem Th3: :: SETFAM_1:3
theorem Th4: :: SETFAM_1:4
for b
1, b
2 being
set holds
( b
1 in b
2 implies
meet b
2 c= b
1 )
theorem Th5: :: SETFAM_1:5
theorem Th6: :: SETFAM_1:6
for b
1, b
2 being
set holds
( b
1 <> {} & ( for b
3 being
set holds
( b
3 in b
1 implies b
2 c= b
3 ) ) implies b
2 c= meet b
1 )
theorem Th7: :: SETFAM_1:7
theorem Th8: :: SETFAM_1:8
for b
1, b
2, b
3 being
set holds
( b
1 in b
2 & b
1 c= b
3 implies
meet b
2 c= b
3 )
theorem Th9: :: SETFAM_1:9
theorem Th10: :: SETFAM_1:10
theorem Th11: :: SETFAM_1:11
theorem Th12: :: SETFAM_1:12
:: deftheorem Def2 defines is_finer_than SETFAM_1:def 2 :
for b
1, b
2 being
set holds
( b
1 is_finer_than b
2 iff for b
3 being
set holds
not ( b
3 in b
1 & ( for b
4 being
set holds
not ( b
4 in b
2 & b
3 c= b
4 ) ) ) );
:: deftheorem Def3 defines is_coarser_than SETFAM_1:def 3 :
for b
1, b
2 being
set holds
( b
2 is_coarser_than b
1 iff for b
3 being
set holds
not ( b
3 in b
2 & ( for b
4 being
set holds
not ( b
4 in b
1 & b
4 c= b
3 ) ) ) );
theorem Th13: :: SETFAM_1:13
canceled;
theorem Th14: :: SETFAM_1:14
canceled;
theorem Th15: :: SETFAM_1:15
canceled;
theorem Th16: :: SETFAM_1:16
canceled;
theorem Th17: :: SETFAM_1:17
theorem Th18: :: SETFAM_1:18
theorem Th19: :: SETFAM_1:19
theorem Th20: :: SETFAM_1:20
theorem Th21: :: SETFAM_1:21
theorem Th22: :: SETFAM_1:22
canceled;
theorem Th23: :: SETFAM_1:23
theorem Th24: :: SETFAM_1:24
theorem Th25: :: SETFAM_1:25
for b
1, b
2, b
3 being
set holds
( b
3 is_finer_than {b1,b2} implies for b
4 being
set holds
not ( b
4 in b
3 & not b
4 c= b
1 & not b
4 c= b
2 ) )
definition
let c
1, c
2 be
set ;
func UNION c
1,c
2 -> set means :
Def4:
:: SETFAM_1:def 4
for b
1 being
set holds
( b
1 in a
3 iff ex b
2, b
3 being
set st
( b
2 in a
1 & b
3 in a
2 & b
1 = b
2 \/ b
3 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being set st
( b3 in c1 & b4 in c2 & b2 = b3 \/ b4 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = b4 \/ b5 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = b4 \/ b5 ) ) ) implies b1 = b2 )
commutativity
for b1 being set
for b2, b3 being set holds
( ( for b4 being set holds
( b4 in b1 iff ex b5, b6 being set st
( b5 in b2 & b6 in b3 & b4 = b5 \/ b6 ) ) ) implies for b4 being set holds
( b4 in b1 iff ex b5, b6 being set st
( b5 in b3 & b6 in b2 & b4 = b5 \/ b6 ) ) )
func INTERSECTION c
1,c
2 -> set means :
Def5:
:: SETFAM_1:def 5
for b
1 being
set holds
( b
1 in a
3 iff ex b
2, b
3 being
set st
( b
2 in a
1 & b
3 in a
2 & b
1 = b
2 /\ b
3 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being set st
( b3 in c1 & b4 in c2 & b2 = b3 /\ b4 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = b4 /\ b5 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = b4 /\ b5 ) ) ) implies b1 = b2 )
commutativity
for b1 being set
for b2, b3 being set holds
( ( for b4 being set holds
( b4 in b1 iff ex b5, b6 being set st
( b5 in b2 & b6 in b3 & b4 = b5 /\ b6 ) ) ) implies for b4 being set holds
( b4 in b1 iff ex b5, b6 being set st
( b5 in b3 & b6 in b2 & b4 = b5 /\ b6 ) ) )
func DIFFERENCE c
1,c
2 -> set means :
Def6:
:: SETFAM_1:def 6
for b
1 being
set holds
( b
1 in a
3 iff ex b
2, b
3 being
set st
( b
2 in a
1 & b
3 in a
2 & b
1 = b
2 \ b
3 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being set st
( b3 in c1 & b4 in c2 & b2 = b3 \ b4 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = b4 \ b5 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = b4 \ b5 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines UNION SETFAM_1:def 4 :
for b
1, b
2 being
set for b
3 being
set holds
( b
3 = UNION b
1,b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5, b
6 being
set st
( b
5 in b
1 & b
6 in b
2 & b
4 = b
5 \/ b
6 ) ) );
:: deftheorem Def5 defines INTERSECTION SETFAM_1:def 5 :
for b
1, b
2 being
set for b
3 being
set holds
( b
3 = INTERSECTION b
1,b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5, b
6 being
set st
( b
5 in b
1 & b
6 in b
2 & b
4 = b
5 /\ b
6 ) ) );
:: deftheorem Def6 defines DIFFERENCE SETFAM_1:def 6 :
for b
1, b
2 being
set for b
3 being
set holds
( b
3 = DIFFERENCE b
1,b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5, b
6 being
set st
( b
5 in b
1 & b
6 in b
2 & b
4 = b
5 \ b
6 ) ) );
theorem Th26: :: SETFAM_1:26
canceled;
theorem Th27: :: SETFAM_1:27
canceled;
theorem Th28: :: SETFAM_1:28
canceled;
theorem Th29: :: SETFAM_1:29
theorem Th30: :: SETFAM_1:30
theorem Th31: :: SETFAM_1:31
theorem Th32: :: SETFAM_1:32
canceled;
theorem Th33: :: SETFAM_1:33
canceled;
theorem Th34: :: SETFAM_1:34
theorem Th35: :: SETFAM_1:35
theorem Th36: :: SETFAM_1:36
theorem Th37: :: SETFAM_1:37
theorem Th38: :: SETFAM_1:38
theorem Th39: :: SETFAM_1:39
theorem Th40: :: SETFAM_1:40
theorem Th41: :: SETFAM_1:41
theorem Th42: :: SETFAM_1:42
canceled;
theorem Th43: :: SETFAM_1:43
canceled;
theorem Th44: :: SETFAM_1:44
for b
1 being
set for b
2, b
3 being
Subset-Family of b
1 holds
( ( for b
4 being
Subset of b
1 holds
( b
4 in b
2 iff b
4 in b
3 ) ) implies b
2 = b
3 )
:: deftheorem Def7 SETFAM_1:def 7 :
canceled;
:: deftheorem Def8 defines COMPLEMENT SETFAM_1:def 8 :
theorem Th45: :: SETFAM_1:45
canceled;
theorem Th46: :: SETFAM_1:46
theorem Th47: :: SETFAM_1:47
theorem Th48: :: SETFAM_1:48
theorem Th49: :: SETFAM_1:49
theorem Th50: :: SETFAM_1:50
canceled;
theorem Th51: :: SETFAM_1:51
theorem Th52: :: SETFAM_1:52
theorem Th53: :: SETFAM_1:53
theorem Th54: :: SETFAM_1:54
theorem Th55: :: SETFAM_1:55
:: deftheorem Def9 defines with_non-empty_elements SETFAM_1:def 9 :
registration
let c
1 be non
empty set ;
cluster {a1} -> with_non-empty_elements ;
coherence
{c1} is with_non-empty_elements
let c
2 be non
empty set ;
cluster {a1,a2} -> with_non-empty_elements ;
coherence
{c1,c2} is with_non-empty_elements
let c
3 be non
empty set ;
cluster {a1,a2,a3} -> with_non-empty_elements ;
coherence
{c1,c2,c3} is with_non-empty_elements
let c
4 be non
empty set ;
cluster {a1,a2,a3,a4} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4} is with_non-empty_elements
let c
5 be non
empty set ;
cluster {a1,a2,a3,a4,a5} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4,c5} is with_non-empty_elements
let c
6 be non
empty set ;
cluster {a1,a2,a3,a4,a5,a6} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4,c5,c6} is with_non-empty_elements
let c
7 be non
empty set ;
cluster {a1,a2,a3,a4,a5,a6,a7} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4,c5,c6,c7} is with_non-empty_elements
let c
8 be non
empty set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4,c5,c6,c7,c8} is with_non-empty_elements
let c
9 be non
empty set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8,a9} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4,c5,c6,c7,c8,c9} is with_non-empty_elements
let c
10 be non
empty set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8,a9,a10} -> with_non-empty_elements ;
coherence
{c1,c2,c3,c4,c5,c6,c7,c8,c9,c10} is with_non-empty_elements
end;
theorem Th56: :: SETFAM_1:56
for b
1, b
2, b
3 being
set holds
(
union b
1 c= b
2 & b
3 in b
1 implies b
3 c= b
2 )
theorem Th57: :: SETFAM_1:57
:: deftheorem Def10 defines Intersect SETFAM_1:def 10 :
theorem Th58: :: SETFAM_1:58
theorem Th59: :: SETFAM_1:59
:: deftheorem Def11 defines empty-membered SETFAM_1:def 11 :