:: ORDINAL3 semantic presentation
theorem Th1: :: ORDINAL3:1
theorem Th2: :: ORDINAL3:2
for b
1, b
2 being
set holds
(
succ b
1 c= b
2 implies b
1 c= b
2 )
theorem Th3: :: ORDINAL3:3
canceled;
theorem Th4: :: ORDINAL3:4
canceled;
theorem Th5: :: ORDINAL3:5
theorem Th6: :: ORDINAL3:6
theorem Th7: :: ORDINAL3:7
theorem Th8: :: ORDINAL3:8
theorem Th9: :: ORDINAL3:9
theorem Th10: :: ORDINAL3:10
theorem Th11: :: ORDINAL3:11
theorem Th12: :: ORDINAL3:12
theorem Th13: :: ORDINAL3:13
theorem Th14: :: ORDINAL3:14
canceled;
theorem Th15: :: ORDINAL3:15
for b
1, b
2 being
Ordinal holds
( b
1 \/ b
2 = b
1 or b
1 \/ b
2 = b
2 )
theorem Th16: :: ORDINAL3:16
for b
1, b
2 being
Ordinal holds
( b
1 /\ b
2 = b
1 or b
1 /\ b
2 = b
2 )
Lemma5:
1 = succ 0
;
theorem Th17: :: ORDINAL3:17
theorem Th18: :: ORDINAL3:18
theorem Th19: :: ORDINAL3:19
for b
1 being
Ordinal holds
not ( b
1 c= 1 & not b
1 = {} & not b
1 = 1 )
theorem Th20: :: ORDINAL3:20
for b
1, b
2, b
3, b
4 being
Ordinal holds
( ( b
1 c= b
2 or b
1 in b
2 ) & b
3 in b
4 implies b
1 +^ b
3 in b
2 +^ b
4 )
theorem Th21: :: ORDINAL3:21
for b
1, b
2, b
3, b
4 being
Ordinal holds
( b
1 c= b
2 & b
3 c= b
4 implies b
1 +^ b
3 c= b
2 +^ b
4 )
theorem Th22: :: ORDINAL3:22
for b
1, b
2, b
3, b
4 being
Ordinal holds
( b
1 in b
2 & ( ( b
3 c= b
4 & b
4 <> {} ) or b
3 in b
4 ) implies b
1 *^ b
3 in b
2 *^ b
4 )
theorem Th23: :: ORDINAL3:23
for b
1, b
2, b
3, b
4 being
Ordinal holds
( b
1 c= b
2 & b
3 c= b
4 implies b
1 *^ b
3 c= b
2 *^ b
4 )
theorem Th24: :: ORDINAL3:24
for b
1, b
2, b
3 being
Ordinal holds
( b
1 +^ b
2 = b
1 +^ b
3 implies b
2 = b
3 )
theorem Th25: :: ORDINAL3:25
for b
1, b
2, b
3 being
Ordinal holds
( b
1 +^ b
2 in b
1 +^ b
3 implies b
2 in b
3 )
theorem Th26: :: ORDINAL3:26
for b
1, b
2, b
3 being
Ordinal holds
( b
1 +^ b
2 c= b
1 +^ b
3 implies b
2 c= b
3 )
theorem Th27: :: ORDINAL3:27
theorem Th28: :: ORDINAL3:28
for b
1, b
2, b
3 being
Ordinal holds
( b
1 in b
2 implies ( b
1 in b
2 +^ b
3 & b
1 in b
3 +^ b
2 ) )
theorem Th29: :: ORDINAL3:29
theorem Th30: :: ORDINAL3:30
for b
1, b
2 being
Ordinal holds
not ( b
1 c= b
2 & ( for b
3 being
Ordinal holds
not b
2 = b
1 +^ b
3 ) )
theorem Th31: :: ORDINAL3:31
theorem Th32: :: ORDINAL3:32
theorem Th33: :: ORDINAL3:33
theorem Th34: :: ORDINAL3:34
theorem Th35: :: ORDINAL3:35
theorem Th36: :: ORDINAL3:36
for b
1, b
2, b
3 being
Ordinal holds
( b
1 *^ b
2 = b
3 *^ b
2 & b
2 <> {} implies b
1 = b
3 )
theorem Th37: :: ORDINAL3:37
for b
1, b
2, b
3 being
Ordinal holds
( b
1 *^ b
2 in b
3 *^ b
2 implies b
1 in b
3 )
theorem Th38: :: ORDINAL3:38
theorem Th39: :: ORDINAL3:39
theorem Th40: :: ORDINAL3:40
canceled;
theorem Th41: :: ORDINAL3:41
for b
1, b
2 being
Ordinal holds
( b
1 *^ b
2 = 1 implies ( b
1 = 1 & b
2 = 1 ) )
theorem Th42: :: ORDINAL3:42
for b
1, b
2, b
3 being
Ordinal holds
not ( b
1 in b
2 +^ b
3 & not b
1 in b
2 & ( for b
4 being
Ordinal holds
not ( b
4 in b
3 & b
1 = b
2 +^ b
4 ) ) )
:: deftheorem Def1 ORDINAL3:def 1 :
canceled;
:: deftheorem Def2 defines +^ ORDINAL3:def 2 :
:: deftheorem Def3 defines +^ ORDINAL3:def 3 :
:: deftheorem Def4 defines *^ ORDINAL3:def 4 :
:: deftheorem Def5 defines *^ ORDINAL3:def 5 :
theorem Th43: :: ORDINAL3:43
canceled;
theorem Th44: :: ORDINAL3:44
canceled;
theorem Th45: :: ORDINAL3:45
canceled;
theorem Th46: :: ORDINAL3:46
canceled;
theorem Th47: :: ORDINAL3:47
theorem Th48: :: ORDINAL3:48
theorem Th49: :: ORDINAL3:49
theorem Th50: :: ORDINAL3:50
theorem Th51: :: ORDINAL3:51
theorem Th52: :: ORDINAL3:52
theorem Th53: :: ORDINAL3:53
theorem Th54: :: ORDINAL3:54
theorem Th55: :: ORDINAL3:55
for b
1, b
2 being
Ordinal holds
not ( b
1 <> {} & ( for b
3, b
4 being
Ordinal holds
not ( b
2 = (b3 *^ b1) +^ b
4 & b
4 in b
1 ) ) )
theorem Th56: :: ORDINAL3:56
for b
1, b
2, b
3, b
4, b
5 being
Ordinal holds
(
(b2 *^ b1) +^ b
3 = (b4 *^ b1) +^ b
5 & b
3 in b
1 & b
5 in b
1 implies ( b
2 = b
4 & b
3 = b
5 ) )
theorem Th57: :: ORDINAL3:57
theorem Th58: :: ORDINAL3:58
:: deftheorem Def6 defines -^ ORDINAL3:def 6 :
for b
1, b
2, b
3 being
Ordinal holds
( ( b
2 c= b
1 implies ( b
3 = b
1 -^ b
2 iff b
1 = b
2 +^ b
3 ) ) & ( not b
2 c= b
1 implies ( b
3 = b
1 -^ b
2 iff b
3 = {} ) ) );
:: deftheorem Def7 defines div^ ORDINAL3:def 7 :
:: deftheorem Def8 defines mod^ ORDINAL3:def 8 :
theorem Th59: :: ORDINAL3:59
canceled;
theorem Th60: :: ORDINAL3:60
for b
1, b
2 being
Ordinal holds
( b
1 in b
2 implies b
2 = b
1 +^ (b2 -^ b1) )
theorem Th61: :: ORDINAL3:61
canceled;
theorem Th62: :: ORDINAL3:62
canceled;
theorem Th63: :: ORDINAL3:63
canceled;
theorem Th64: :: ORDINAL3:64
canceled;
theorem Th65: :: ORDINAL3:65
theorem Th66: :: ORDINAL3:66
for b
1, b
2, b
3 being
Ordinal holds
( b
1 in b
2 & ( b
3 c= b
1 or b
3 in b
1 ) implies b
1 -^ b
3 in b
2 -^ b
3 )
theorem Th67: :: ORDINAL3:67
theorem Th68: :: ORDINAL3:68
theorem Th69: :: ORDINAL3:69
theorem Th70: :: ORDINAL3:70
theorem Th71: :: ORDINAL3:71
for b
1, b
2, b
3 being
Ordinal holds
( b
1 c= b
2 implies b
3 -^ b
2 c= b
3 -^ b
1 )
theorem Th72: :: ORDINAL3:72
for b
1, b
2, b
3 being
Ordinal holds
( b
1 c= b
2 implies b
1 -^ b
3 c= b
2 -^ b
3 )
theorem Th73: :: ORDINAL3:73
theorem Th74: :: ORDINAL3:74
for b
1, b
2, b
3 being
Ordinal holds
( b
1 +^ b
2 in b
3 implies b
2 in b
3 -^ b
1 )
theorem Th75: :: ORDINAL3:75
theorem Th76: :: ORDINAL3:76
theorem Th77: :: ORDINAL3:77
theorem Th78: :: ORDINAL3:78
theorem Th79: :: ORDINAL3:79
for b
1, b
2, b
3, b
4 being
Ordinal holds
( b
1 = (b2 *^ b3) +^ b
4 & b
4 in b
3 implies ( b
2 = b
1 div^ b
3 & b
4 = b
1 mod^ b
3 ) )
theorem Th80: :: ORDINAL3:80
theorem Th81: :: ORDINAL3:81
theorem Th82: :: ORDINAL3:82
theorem Th83: :: ORDINAL3:83
theorem Th84: :: ORDINAL3:84