:: CARD_3 semantic presentation
:: deftheorem Def1 defines Cardinal-yielding CARD_3:def 1 :
theorem Th1: :: CARD_3:1
canceled;
theorem Th2: :: CARD_3:2
canceled;
theorem Th3: :: CARD_3:3
:: deftheorem Def2 defines Card CARD_3:def 2 :
:: deftheorem Def3 defines disjoin CARD_3:def 3 :
:: deftheorem Def4 defines Union CARD_3:def 4 :
:: deftheorem Def5 defines product CARD_3:def 5 :
theorem Th4: :: CARD_3:4
canceled;
theorem Th5: :: CARD_3:5
canceled;
theorem Th6: :: CARD_3:6
canceled;
theorem Th7: :: CARD_3:7
canceled;
theorem Th8: :: CARD_3:8
theorem Th9: :: CARD_3:9
theorem Th10: :: CARD_3:10
theorem Th11: :: CARD_3:11
theorem Th12: :: CARD_3:12
theorem Th13: :: CARD_3:13
theorem Th14: :: CARD_3:14
theorem Th15: :: CARD_3:15
theorem Th16: :: CARD_3:16
theorem Th17: :: CARD_3:17
theorem Th18: :: CARD_3:18
theorem Th19: :: CARD_3:19
theorem Th20: :: CARD_3:20
defpred S1[ set ] means a1 is Function;
:: deftheorem Def6 defines pi CARD_3:def 6 :
for b
1, b
2, b
3 being
set holds
( b
3 = pi b
2,b
1 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5 being
Function st
( b
5 in b
2 & b
4 = b
5 . b
1 ) ) );
theorem Th21: :: CARD_3:21
canceled;
theorem Th22: :: CARD_3:22
theorem Th23: :: CARD_3:23
canceled;
theorem Th24: :: CARD_3:24
theorem Th25: :: CARD_3:25
theorem Th26: :: CARD_3:26
theorem Th27: :: CARD_3:27
for b
1, b
2, b
3 being
set holds
pi (b1 \/ b2),b
3 = (pi b1,b3) \/ (pi b2,b3)
theorem Th28: :: CARD_3:28
for b
1, b
2, b
3 being
set holds
pi (b1 /\ b2),b
3 c= (pi b1,b3) /\ (pi b2,b3)
theorem Th29: :: CARD_3:29
for b
1, b
2, b
3 being
set holds
(pi b1,b2) \ (pi b3,b2) c= pi (b1 \ b3),b
2
theorem Th30: :: CARD_3:30
theorem Th31: :: CARD_3:31
theorem Th32: :: CARD_3:32
theorem Th33: :: CARD_3:33
theorem Th34: :: CARD_3:34
theorem Th35: :: CARD_3:35
theorem Th36: :: CARD_3:36
theorem Th37: :: CARD_3:37
theorem Th38: :: CARD_3:38
theorem Th39: :: CARD_3:39
theorem Th40: :: CARD_3:40
:: deftheorem Def7 defines Sum CARD_3:def 7 :
:: deftheorem Def8 defines Product CARD_3:def 8 :
theorem Th41: :: CARD_3:41
canceled;
theorem Th42: :: CARD_3:42
canceled;
theorem Th43: :: CARD_3:43
theorem Th44: :: CARD_3:44
theorem Th45: :: CARD_3:45
theorem Th46: :: CARD_3:46
theorem Th47: :: CARD_3:47
theorem Th48: :: CARD_3:48
theorem Th49: :: CARD_3:49
theorem Th50: :: CARD_3:50
theorem Th51: :: CARD_3:51
theorem Th52: :: CARD_3:52
theorem Th53: :: CARD_3:53
theorem Th54: :: CARD_3:54
theorem Th55: :: CARD_3:55
theorem Th56: :: CARD_3:56
scheme :: CARD_3:sch 2
s2{ F
1()
-> finite set , P
1[
set ,
set ] } :
ex b
1 being
set st
( b
1 in F
1() & ( for b
2 being
set holds
not ( b
2 in F
1() & b
2 <> b
1 & P
1[b
2,b
1] ) ) )
provided
E25:
F
1()
<> {}
and
E26:
for b
1, b
2 being
set holds
( P
1[b
1,b
2] & P
1[b
2,b
1] implies b
1 = b
2 )
and
E27:
for b
1, b
2, b
3 being
set holds
( P
1[b
1,b
2] & P
1[b
2,b
3] implies P
1[b
1,b
3] )
scheme :: CARD_3:sch 3
s3{ F
1()
-> finite set , P
1[
set ,
set ] } :
ex b
1 being
set st
( b
1 in F
1() & ( for b
2 being
set holds
( b
2 in F
1() implies P
1[b
1,b
2] ) ) )
provided
E25:
F
1()
<> {}
and
E26:
for b
1, b
2 being
set holds
( P
1[b
1,b
2] or P
1[b
2,b
1] )
and
E27:
for b
1, b
2, b
3 being
set holds
( P
1[b
1,b
2] & P
1[b
2,b
3] implies P
1[b
1,b
3] )
Lemma25:
for b1 being Nat holds
( Rank b1 is finite implies Rank (b1 + 1) is finite )
theorem Th57: :: CARD_3:57
Lemma26:
for b1 being set
for b2 being Relation holds
not ( b1 in field b2 & ( for b3 being set holds
( not [b1,b3] in b2 & not [b3,b1] in b2 ) ) )
theorem Th58: :: CARD_3:58
theorem Th59: :: CARD_3:59
theorem Th60: :: CARD_3:60
theorem Th61: :: CARD_3:61
for b
1 being
set holds
not ( b
1 is
c=-linear & ( for b
2 being
set holds
not ( b
2 c= b
1 &
union b
2 = union b
1 & ( for b
3 being
set holds
not ( b
3 c= b
2 & b
3 <> {} & ( for b
4 being
set holds
not ( b
4 in b
3 & ( for b
5 being
set holds
( b
5 in b
3 implies b
4 c= b
5 ) ) ) ) ) ) ) ) )
theorem Th62: :: CARD_3:62
theorem Th63: :: CARD_3:63
theorem Th64: :: CARD_3:64