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Theorem fin23lem24 8194
Description: Lemma for fin23 8261. 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 731 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  A )
2 simplr 732 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  B  C_  A )
3 simprl 733 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  B )
42, 3sseldd 3341 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  A )
5 ordelord 4595 . . . . . 6  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
61, 4, 5syl2anc 643 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  C )
7 simprr 734 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  B )
82, 7sseldd 3341 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  A )
9 ordelord 4595 . . . . . 6  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
101, 8, 9syl2anc 643 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  D )
11 ordtri3 4609 . . . . . 6  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  =  D  <->  -.  ( C  e.  D  \/  D  e.  C ) ) )
1211necon2abid 2655 . . . . 5  |-  ( ( Ord  C  /\  Ord  D )  ->  ( ( C  e.  D  \/  D  e.  C )  <->  C  =/=  D ) )
136, 10, 12syl2anc 643 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  e.  D  \/  D  e.  C
)  <->  C  =/=  D
) )
14 simpr 448 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  D )
15 simplrl 737 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  B )
16 elin 3522 . . . . . . . . 9  |-  ( C  e.  ( D  i^i  B )  <->  ( C  e.  D  /\  C  e.  B ) )
1714, 15, 16sylanbrc 646 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  ( D  i^i  B ) )
186adantr 452 . . . . . . . . . 10  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  Ord  C )
19 ordirr 4591 . . . . . . . . . 10  |-  ( Ord 
C  ->  -.  C  e.  C )
2018, 19syl 16 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  C
)
21 inss1 3553 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  C
2221sseli 3336 . . . . . . . . 9  |-  ( C  e.  ( C  i^i  B )  ->  C  e.  C )
2320, 22nsyl 115 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  ( C  i^i  B ) )
24 nelne1 2687 . . . . . . . 8  |-  ( ( C  e.  ( D  i^i  B )  /\  -.  C  e.  ( C  i^i  B ) )  ->  ( D  i^i  B )  =/=  ( C  i^i  B ) )
2517, 23, 24syl2anc 643 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  ( D  i^i  B
)  =/=  ( C  i^i  B ) )
2625necomd 2681 . . . . . 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 448 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  C )
28 simplrr 738 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  B )
29 elin 3522 . . . . . . . 8  |-  ( D  e.  ( C  i^i  B )  <->  ( D  e.  C  /\  D  e.  B ) )
3027, 28, 29sylanbrc 646 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  ( C  i^i  B ) )
3110adantr 452 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  Ord  D )
32 ordirr 4591 . . . . . . . . 9  |-  ( Ord 
D  ->  -.  D  e.  D )
3331, 32syl 16 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  D
)
34 inss1 3553 . . . . . . . . 9  |-  ( D  i^i  B )  C_  D
3534sseli 3336 . . . . . . . 8  |-  ( D  e.  ( D  i^i  B )  ->  D  e.  D )
3633, 35nsyl 115 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  ( D  i^i  B ) )
37 nelne1 2687 . . . . . . 7  |-  ( ( D  e.  ( C  i^i  B )  /\  -.  D  e.  ( D  i^i  B ) )  ->  ( C  i^i  B )  =/=  ( D  i^i  B ) )
3830, 36, 37syl2anc 643 . . . . . 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 761 . . . . 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 424 . . . 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 227 . . 3  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( C  =/=  D  ->  ( C  i^i  B )  =/=  ( D  i^i  B
) ) )
4241necon4d 2661 . 2  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  ->  C  =  D )
)
43 ineq1 3527 . 2  |-  ( C  =  D  ->  ( C  i^i  B )  =  ( D  i^i  B
) )
4442, 43impbid1 195 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 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311    C_ wss 3312   Ord word 4572
This theorem is referenced by:  fin23lem23  8198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576
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