:: WELLORD2 semantic presentation
:: deftheorem Def1 defines RelIncl WELLORD2:def 1 :
theorem Th1: :: WELLORD2:1
canceled;
theorem Th2: :: WELLORD2:2
theorem Th3: :: WELLORD2:3
theorem Th4: :: WELLORD2:4
theorem Th5: :: WELLORD2:5
theorem Th6: :: WELLORD2:6
theorem Th7: :: WELLORD2:7
theorem Th8: :: WELLORD2:8
theorem Th9: :: WELLORD2:9
theorem Th10: :: WELLORD2:10
theorem Th11: :: WELLORD2:11
theorem Th12: :: WELLORD2:12
theorem Th13: :: WELLORD2:13
theorem Th14: :: WELLORD2:14
:: deftheorem Def2 defines order_type_of WELLORD2:def 2 :
:: deftheorem Def3 defines is_order_type_of WELLORD2:def 3 :
theorem Th15: :: WELLORD2:15
canceled;
theorem Th16: :: WELLORD2:16
canceled;
theorem Th17: :: WELLORD2:17
:: deftheorem Def4 defines are_equipotent WELLORD2:def 4 :
theorem Th18: :: WELLORD2:18
canceled;
theorem Th19: :: WELLORD2:19
canceled;
theorem Th20: :: WELLORD2:20
canceled;
theorem Th21: :: WELLORD2:21
canceled;
theorem Th22: :: WELLORD2:22
theorem Th23: :: WELLORD2:23
canceled;
theorem Th24: :: WELLORD2:24
canceled;
theorem Th25: :: WELLORD2:25
Lemma17:
for b1 being set
for b2 being Relation holds
not ( b2 is well-ordering & b1, field b2 are_equipotent & ( for b3 being Relation holds
not b3 well_orders b1 ) )
theorem Th26: :: WELLORD2:26
theorem Th27: :: WELLORD2:27
for b
1 being non
empty set holds
not ( ( for b
2 being
set holds
not ( b
2 in b
1 & not b
2 <> {} ) ) & ( for b
2, b
3 being
set holds
( b
2 in b
1 & b
3 in b
1 & b
2 <> b
3 implies b
2 misses b
3 ) ) & ( for b
2 being
set holds
ex b
3 being
set st
( b
3 in b
1 & ( for b
4 being
set holds
not b
2 /\ b
3 = {b4} ) ) ) )
theorem Th28: :: WELLORD2:28
for b
1 being non
empty set holds
not ( ( for b
2 being
set holds
not ( b
2 in b
1 & not b
2 <> {} ) ) & ( for b
2 being
Function holds
not (
dom b
2 = b
1 & ( for b
3 being
set holds
( b
3 in b
1 implies b
2 . b
3 in b
3 ) ) ) ) )
scheme :: WELLORD2:sch 2
s2{ F
1()
-> set , F
2()
-> set , F
3()
-> set , P
1[
set ,
set ,
set ] } :
ex b
1, b
2 being
Function st
(
dom b
1 = F
1() &
dom b
2 = F
1() & ( for b
3 being
set holds
( b
3 in F
1() implies P
1[b
3,b
1 . b
3,b
2 . b
3] ) ) )
provided
E20:
for b
1 being
set holds
not ( b
1 in F
1() & ( for b
2, b
3 being
set holds
not ( b
2 in F
2() & b
3 in F
3() & P
1[b
1,b
2,b
3] ) ) )
theorem Th29: :: WELLORD2:29
theorem Th30: :: WELLORD2:30
theorem Th31: :: WELLORD2:31