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Theorem fin23lem24 8490
Description: Lemma for fin23 8557. 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 753 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  A )
2 simplr 754 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  B  C_  A )
3 simprl 755 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  B )
42, 3sseldd 3356 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  A )
5 ordelord 4740 . . . . . 6  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
61, 4, 5syl2anc 661 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  C )
7 simprr 756 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  B )
82, 7sseldd 3356 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  A )
9 ordelord 4740 . . . . . 6  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
101, 8, 9syl2anc 661 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  D )
11 ordtri3 4754 . . . . . 6  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  =  D  <->  -.  ( C  e.  D  \/  D  e.  C ) ) )
1211necon2abid 2667 . . . . 5  |-  ( ( Ord  C  /\  Ord  D )  ->  ( ( C  e.  D  \/  D  e.  C )  <->  C  =/=  D ) )
136, 10, 12syl2anc 661 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  e.  D  \/  D  e.  C
)  <->  C  =/=  D
) )
14 simpr 461 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  D )
15 simplrl 759 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  B )
1614, 15elind 3539 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  ( D  i^i  B ) )
176adantr 465 . . . . . . . . . 10  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  Ord  C )
18 ordirr 4736 . . . . . . . . . 10  |-  ( Ord 
C  ->  -.  C  e.  C )
1917, 18syl 16 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  C
)
20 inss1 3569 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  C
2120sseli 3351 . . . . . . . . 9  |-  ( C  e.  ( C  i^i  B )  ->  C  e.  C )
2219, 21nsyl 121 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  ( C  i^i  B ) )
23 nelne1 2700 . . . . . . . 8  |-  ( ( C  e.  ( D  i^i  B )  /\  -.  C  e.  ( C  i^i  B ) )  ->  ( D  i^i  B )  =/=  ( C  i^i  B ) )
2416, 22, 23syl2anc 661 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  ( D  i^i  B
)  =/=  ( C  i^i  B ) )
2524necomd 2694 . . . . . 6  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  ( C  i^i  B
)  =/=  ( D  i^i  B ) )
26 simpr 461 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  C )
27 simplrr 760 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  B )
2826, 27elind 3539 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  ( C  i^i  B ) )
2910adantr 465 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  Ord  D )
30 ordirr 4736 . . . . . . . . 9  |-  ( Ord 
D  ->  -.  D  e.  D )
3129, 30syl 16 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  D
)
32 inss1 3569 . . . . . . . . 9  |-  ( D  i^i  B )  C_  D
3332sseli 3351 . . . . . . . 8  |-  ( D  e.  ( D  i^i  B )  ->  D  e.  D )
3431, 33nsyl 121 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  ( D  i^i  B ) )
35 nelne1 2700 . . . . . . 7  |-  ( ( D  e.  ( C  i^i  B )  /\  -.  D  e.  ( D  i^i  B ) )  ->  ( C  i^i  B )  =/=  ( D  i^i  B ) )
3628, 34, 35syl2anc 661 . . . . . 6  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  ( C  i^i  B
)  =/=  ( D  i^i  B ) )
3725, 36jaodan 783 . . . . 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
) )
3837ex 434 . . . 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 ) ) )
3913, 38sylbird 235 . . 3  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( C  =/=  D  ->  ( C  i^i  B )  =/=  ( D  i^i  B
) ) )
4039necon4d 2673 . 2  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  ->  C  =  D )
)
41 ineq1 3544 . 2  |-  ( C  =  D  ->  ( C  i^i  B )  =  ( D  i^i  B
) )
4240, 41impbid1 203 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605    i^i cin 3326    C_ wss 3327   Ord word 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-tr 4385  df-eprel 4631  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721
This theorem is referenced by:  fin23lem23  8494
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