:: CARD_4 semantic presentation
Lemma1:
( 0 = Card 0 & 1 = Card 1 & 2 = Card 2 )
by CARD_1:def 5;
theorem Th1: :: CARD_4:1
theorem Th2: :: CARD_4:2
theorem Th3: :: CARD_4:3
theorem Th4: :: CARD_4:4
theorem Th5: :: CARD_4:5
theorem Th6: :: CARD_4:6
theorem Th7: :: CARD_4:7
theorem Th8: :: CARD_4:8
theorem Th9: :: CARD_4:9
theorem Th10: :: CARD_4:10
canceled;
theorem Th11: :: CARD_4:11
theorem Th12: :: CARD_4:12
canceled;
theorem Th13: :: CARD_4:13
theorem Th14: :: CARD_4:14
theorem Th15: :: CARD_4:15
theorem Th16: :: CARD_4:16
canceled;
theorem Th17: :: CARD_4:17
theorem Th18: :: CARD_4:18
canceled;
theorem Th19: :: CARD_4:19
theorem Th20: :: CARD_4:20
theorem Th21: :: CARD_4:21
theorem Th22: :: CARD_4:22
theorem Th23: :: CARD_4:23
theorem Th24: :: CARD_4:24
theorem Th25: :: CARD_4:25
theorem Th26: :: CARD_4:26
set c1 = succ 1;
theorem Th27: :: CARD_4:27
theorem Th28: :: CARD_4:28
theorem Th29: :: CARD_4:29
theorem Th30: :: CARD_4:30
theorem Th31: :: CARD_4:31
theorem Th32: :: CARD_4:32
theorem Th33: :: CARD_4:33
theorem Th34: :: CARD_4:34
theorem Th35: :: CARD_4:35
theorem Th36: :: CARD_4:36
theorem Th37: :: CARD_4:37
theorem Th38: :: CARD_4:38
theorem Th39: :: CARD_4:39
theorem Th40: :: CARD_4:40
theorem Th41: :: CARD_4:41
for b
1, b
2, b
3, b
4 being
Cardinal holds
( not ( not ( b
1 <` b
2 & b
3 <` b
4 ) & not ( b
1 <=` b
2 & b
3 <` b
4 ) & not ( b
1 <` b
2 & b
3 <=` b
4 ) & not ( b
1 <=` b
2 & b
3 <=` b
4 ) ) implies ( b
1 +` b
3 <=` b
2 +` b
4 & b
3 +` b
1 <=` b
2 +` b
4 ) )
theorem Th42: :: CARD_4:42
:: deftheorem Def1 defines countable CARD_4:def 1 :
theorem Th43: :: CARD_4:43
theorem Th44: :: CARD_4:44
theorem Th45: :: CARD_4:45
theorem Th46: :: CARD_4:46
theorem Th47: :: CARD_4:47
theorem Th48: :: CARD_4:48
theorem Th49: :: CARD_4:49
theorem Th50: :: CARD_4:50
theorem Th51: :: CARD_4:51
for b
1 being
Natfor b
2 being
Real holds
( not ( not ( not b
2 <> 0 & not b
1 = 0 ) & not b
2 |^ b
1 <> 0 ) & not ( b
2 |^ b
1 <> 0 & not b
2 <> 0 & not b
1 = 0 ) )
Lemma32:
for b1, b2, b3, b4 being Nat holds
( (2 |^ b1) * ((2 * b2) + 1) = (2 |^ b3) * ((2 * b4) + 1) implies b1 <= b3 )
theorem Th52: :: CARD_4:52
for b
1, b
2, b
3, b
4 being
Nat holds
(
(2 |^ b1) * ((2 * b2) + 1) = (2 |^ b3) * ((2 * b4) + 1) implies ( b
1 = b
3 & b
2 = b
4 ) )
Lemma34:
for b1 being set holds
not ( b1 in [:NAT ,NAT :] & ( for b2, b3 being Nat holds
not b1 = [b2,b3] ) )
theorem Th53: :: CARD_4:53
theorem Th54: :: CARD_4:54
theorem Th55: :: CARD_4:55
theorem Th56: :: CARD_4:56
theorem Th57: :: CARD_4:57
theorem Th58: :: CARD_4:58
theorem Th59: :: CARD_4:59
theorem Th60: :: CARD_4:60
theorem Th61: :: CARD_4:61
theorem Th62: :: CARD_4:62
theorem Th63: :: CARD_4:63
theorem Th64: :: CARD_4:64
canceled;
theorem Th65: :: CARD_4:65
theorem Th66: :: CARD_4:66
theorem Th67: :: CARD_4:67
theorem Th68: :: CARD_4:68
for b
1, b
2, b
3, b
4 being
Cardinal holds
( not ( not ( b
1 <` b
2 & b
3 <` b
4 ) & not ( b
1 <=` b
2 & b
3 <` b
4 ) & not ( b
1 <` b
2 & b
3 <=` b
4 ) & not ( b
1 <=` b
2 & b
3 <=` b
4 ) ) implies ( b
1 *` b
3 <=` b
2 *` b
4 & b
3 *` b
1 <=` b
2 *` b
4 ) )
theorem Th69: :: CARD_4:69
theorem Th70: :: CARD_4:70
for b
1, b
2, b
3, b
4 being
Cardinal holds
not ( not ( not ( b
1 <` b
2 & b
3 <` b
4 ) & not ( b
1 <=` b
2 & b
3 <` b
4 ) & not ( b
1 <` b
2 & b
3 <=` b
4 ) & not ( b
1 <=` b
2 & b
3 <=` b
4 ) ) & not b
1 = 0 & not
exp b
1,b
3 <=` exp b
2,b
4 )
theorem Th71: :: CARD_4:71
theorem Th72: :: CARD_4:72
theorem Th73: :: CARD_4:73
theorem Th74: :: CARD_4:74
theorem Th75: :: CARD_4:75
theorem Th76: :: CARD_4:76
theorem Th77: :: CARD_4:77
theorem Th78: :: CARD_4:78
theorem Th79: :: CARD_4:79
theorem Th80: :: CARD_4:80
theorem Th81: :: CARD_4:81
theorem Th82: :: CARD_4:82
theorem Th83: :: CARD_4:83
theorem Th84: :: CARD_4:84
theorem Th85: :: CARD_4:85
theorem Th86: :: CARD_4:86
theorem Th87: :: CARD_4:87