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Theorem fin23lem24 7964
Description: Lemma for fin23 8031. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Assertion
Ref Expression
fin23lem24  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  <->  C  =  D ) )

Proof of Theorem fin23lem24
StepHypRef Expression
1 simpll 730 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  A )
2 simplr 731 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  B  C_  A )
3 simprl 732 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  B )
42, 3sseldd 3194 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  A )
5 ordelord 4430 . . . . . 6  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
61, 4, 5syl2anc 642 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  C )
7 simprr 733 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  B )
82, 7sseldd 3194 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  A )
9 ordelord 4430 . . . . . 6  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
101, 8, 9syl2anc 642 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  D )
11 ordtri3 4444 . . . . . 6  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  =  D  <->  -.  ( C  e.  D  \/  D  e.  C ) ) )
1211necon2abid 2516 . . . . 5  |-  ( ( Ord  C  /\  Ord  D )  ->  ( ( C  e.  D  \/  D  e.  C )  <->  C  =/=  D ) )
136, 10, 12syl2anc 642 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  e.  D  \/  D  e.  C
)  <->  C  =/=  D
) )
14 simpr 447 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  D )
15 simplrl 736 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  B )
16 elin 3371 . . . . . . . . 9  |-  ( C  e.  ( D  i^i  B )  <->  ( C  e.  D  /\  C  e.  B ) )
1714, 15, 16sylanbrc 645 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  ( D  i^i  B ) )
186adantr 451 . . . . . . . . . 10  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  Ord  C )
19 ordirr 4426 . . . . . . . . . 10  |-  ( Ord 
C  ->  -.  C  e.  C )
2018, 19syl 15 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  C
)
21 inss1 3402 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  C
2221sseli 3189 . . . . . . . . 9  |-  ( C  e.  ( C  i^i  B )  ->  C  e.  C )
2320, 22nsyl 113 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  ( C  i^i  B ) )
24 nelne1 2548 . . . . . . . 8  |-  ( ( C  e.  ( D  i^i  B )  /\  -.  C  e.  ( C  i^i  B ) )  ->  ( D  i^i  B )  =/=  ( C  i^i  B ) )
2517, 23, 24syl2anc 642 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  ( D  i^i  B
)  =/=  ( C  i^i  B ) )
2625necomd 2542 . . . . . 6  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  ( C  i^i  B
)  =/=  ( D  i^i  B ) )
27 simpr 447 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  C )
28 simplrr 737 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  B )
29 elin 3371 . . . . . . . 8  |-  ( D  e.  ( C  i^i  B )  <->  ( D  e.  C  /\  D  e.  B ) )
3027, 28, 29sylanbrc 645 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  ( C  i^i  B ) )
3110adantr 451 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  Ord  D )
32 ordirr 4426 . . . . . . . . 9  |-  ( Ord 
D  ->  -.  D  e.  D )
3331, 32syl 15 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  D
)
34 inss1 3402 . . . . . . . . 9  |-  ( D  i^i  B )  C_  D
3534sseli 3189 . . . . . . . 8  |-  ( D  e.  ( D  i^i  B )  ->  D  e.  D )
3633, 35nsyl 113 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  ( D  i^i  B ) )
37 nelne1 2548 . . . . . . 7  |-  ( ( D  e.  ( C  i^i  B )  /\  -.  D  e.  ( D  i^i  B ) )  ->  ( C  i^i  B )  =/=  ( D  i^i  B ) )
3830, 36, 37syl2anc 642 . . . . . 6  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  ( C  i^i  B
)  =/=  ( D  i^i  B ) )
3926, 38jaodan 760 . . . . 5  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  ( C  e.  D  \/  D  e.  C
) )  ->  ( C  i^i  B )  =/=  ( D  i^i  B
) )
4039ex 423 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  e.  D  \/  D  e.  C
)  ->  ( C  i^i  B )  =/=  ( D  i^i  B ) ) )
4113, 40sylbird 226 . . 3  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( C  =/=  D  ->  ( C  i^i  B )  =/=  ( D  i^i  B
) ) )
4241necon4d 2522 . 2  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  ->  C  =  D )
)
43 ineq1 3376 . 2  |-  ( C  =  D  ->  ( C  i^i  B )  =  ( D  i^i  B
) )
4442, 43impbid1 194 1  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  <->  C  =  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   Ord word 4407
This theorem is referenced by:  fin23lem23  7968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411
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