:: ORDINAL1 semantic presentation
theorem Th1: :: ORDINAL1:1
canceled;
theorem Th2: :: ORDINAL1:2
canceled;
theorem Th3: :: ORDINAL1:3
for b
1, b
2, b
3 being
set holds
not ( b
1 in b
2 & b
2 in b
3 & b
3 in b
1 )
theorem Th4: :: ORDINAL1:4
for b
1, b
2, b
3, b
4 being
set holds
not ( b
1 in b
2 & b
2 in b
3 & b
3 in b
4 & b
4 in b
1 )
theorem Th5: :: ORDINAL1:5
for b
1, b
2, b
3, b
4, b
5 being
set holds
not ( b
1 in b
2 & b
2 in b
3 & b
3 in b
4 & b
4 in b
5 & b
5 in b
1 )
theorem Th6: :: ORDINAL1:6
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
not ( b
1 in b
2 & b
2 in b
3 & b
3 in b
4 & b
4 in b
5 & b
5 in b
6 & b
6 in b
1 )
theorem Th7: :: ORDINAL1:7
for b
1, b
2 being
set holds
not ( b
1 in b
2 & b
2 c= b
1 )
:: deftheorem Def1 defines succ ORDINAL1:def 1 :
theorem Th8: :: ORDINAL1:8
canceled;
theorem Th9: :: ORDINAL1:9
canceled;
theorem Th10: :: ORDINAL1:10
theorem Th11: :: ORDINAL1:11
canceled;
theorem Th12: :: ORDINAL1:12
theorem Th13: :: ORDINAL1:13
for b
1, b
2 being
set holds
( b
1 in succ b
2 iff ( b
1 in b
2 or b
1 = b
2 ) )
theorem Th14: :: ORDINAL1:14
:: deftheorem Def2 defines epsilon-transitive ORDINAL1:def 2 :
:: deftheorem Def3 defines epsilon-connected ORDINAL1:def 3 :
Lemma8:
( {} is epsilon-transitive & {} is epsilon-connected )
:: deftheorem Def4 defines ordinal ORDINAL1:def 4 :
theorem Th15: :: ORDINAL1:15
canceled;
theorem Th16: :: ORDINAL1:16
canceled;
theorem Th17: :: ORDINAL1:17
canceled;
theorem Th18: :: ORDINAL1:18
canceled;
theorem Th19: :: ORDINAL1:19
theorem Th20: :: ORDINAL1:20
canceled;
theorem Th21: :: ORDINAL1:21
theorem Th22: :: ORDINAL1:22
theorem Th23: :: ORDINAL1:23
theorem Th24: :: ORDINAL1:24
for b
1, b
2 being
Ordinal holds
not ( not b
1 in b
2 & not b
1 = b
2 & not b
2 in b
1 )
theorem Th25: :: ORDINAL1:25
theorem Th26: :: ORDINAL1:26
for b
1, b
2 being
Ordinal holds
( b
1 c= b
2 or b
2 in b
1 )
theorem Th27: :: ORDINAL1:27
theorem Th28: :: ORDINAL1:28
canceled;
theorem Th29: :: ORDINAL1:29
theorem Th30: :: ORDINAL1:30
theorem Th31: :: ORDINAL1:31
for b
1 being
set holds
( ( for b
2 being
set holds
( b
2 in b
1 implies ( b
2 is
Ordinal & b
2 c= b
1 ) ) ) implies b
1 is
ordinal )
theorem Th32: :: ORDINAL1:32
for b
1 being
set for b
2 being
Ordinal holds
not ( b
1 c= b
2 & b
1 <> {} & ( for b
3 being
Ordinal holds
not ( b
3 in b
1 & ( for b
4 being
Ordinal holds
( b
4 in b
1 implies b
3 c= b
4 ) ) ) ) )
theorem Th33: :: ORDINAL1:33
theorem Th34: :: ORDINAL1:34
theorem Th35: :: ORDINAL1:35
theorem Th36: :: ORDINAL1:36
for b
1 being
set holds
not ( ( for b
2 being
set holds
( b
2 in b
1 implies b
2 is
Ordinal ) ) & ( for b
2 being
Ordinal holds
not b
1 c= b
2 ) )
theorem Th37: :: ORDINAL1:37
for b
1 being
set holds
not for b
2 being
set holds
( b
2 in b
1 iff b
2 is
Ordinal )
theorem Th38: :: ORDINAL1:38
for b
1 being
set holds
not for b
2 being
Ordinal holds b
2 in b
1
theorem Th39: :: ORDINAL1:39
for b
1 being
set holds
ex b
2 being
Ordinal st
( not b
2 in b
1 & ( for b
3 being
Ordinal holds
( not b
3 in b
1 implies b
2 c= b
3 ) ) )
:: deftheorem Def5 ORDINAL1:def 5 :
canceled;
:: deftheorem Def6 defines being_limit_ordinal ORDINAL1:def 6 :
theorem Th40: :: ORDINAL1:40
canceled;
theorem Th41: :: ORDINAL1:41
theorem Th42: :: ORDINAL1:42
:: deftheorem Def7 defines T-Sequence-like ORDINAL1:def 7 :
:: deftheorem Def8 defines T-Sequence ORDINAL1:def 8 :
theorem Th43: :: ORDINAL1:43
canceled;
theorem Th44: :: ORDINAL1:44
canceled;
theorem Th45: :: ORDINAL1:45
theorem Th46: :: ORDINAL1:46
theorem Th47: :: ORDINAL1:47
theorem Th48: :: ORDINAL1:48
:: deftheorem Def9 defines c=-linear ORDINAL1:def 9 :
theorem Th49: :: ORDINAL1:49
theorem Th50: :: ORDINAL1:50
for b
1, b
2 being
Ordinal holds
not ( not b
1 c< b
2 & not b
1 = b
2 & not b
2 c< b
1 )
:: deftheorem Def10 defines On ORDINAL1:def 10 :
for b
1, b
2 being
set holds
( b
2 = On b
1 iff for b
3 being
set holds
( b
3 in b
2 iff ( b
3 in b
1 & b
3 is
Ordinal ) ) );
:: deftheorem Def11 defines Lim ORDINAL1:def 11 :
theorem Th51: :: ORDINAL1:51
:: deftheorem Def12 defines omega ORDINAL1:def 12 :
:: deftheorem Def13 defines natural ORDINAL1:def 13 :