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Theorem fin23lem24 8750
Description: Lemma for fin23 8817. 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 758 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  A )
2 simplr 760 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  B  C_  A )
3 simprl 762 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  B )
42, 3sseldd 3471 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  A )
5 ordelord 5464 . . . . . 6  |-  ( ( Ord  A  /\  C  e.  A )  ->  Ord  C )
61, 4, 5syl2anc 665 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  C )
7 simprr 764 . . . . . . 7  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  B )
82, 7sseldd 3471 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  A )
9 ordelord 5464 . . . . . 6  |-  ( ( Ord  A  /\  D  e.  A )  ->  Ord  D )
101, 8, 9syl2anc 665 . . . . 5  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  Ord  D )
11 ordtri3 5478 . . . . . 6  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  =  D  <->  -.  ( C  e.  D  \/  D  e.  C ) ) )
1211necon2abid 2685 . . . . 5  |-  ( ( Ord  C  /\  Ord  D )  ->  ( ( C  e.  D  \/  D  e.  C )  <->  C  =/=  D ) )
136, 10, 12syl2anc 665 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  e.  D  \/  D  e.  C
)  <->  C  =/=  D
) )
14 simpr 462 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  D )
15 simplrl 768 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  B )
1614, 15elind 3656 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  C  e.  ( D  i^i  B ) )
176adantr 466 . . . . . . . . . 10  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  Ord  C )
18 ordirr 5460 . . . . . . . . . 10  |-  ( Ord 
C  ->  -.  C  e.  C )
1917, 18syl 17 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  C
)
20 inss1 3688 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  C
2120sseli 3466 . . . . . . . . 9  |-  ( C  e.  ( C  i^i  B )  ->  C  e.  C )
2219, 21nsyl 124 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  -.  C  e.  ( C  i^i  B ) )
23 nelne1 2760 . . . . . . . 8  |-  ( ( C  e.  ( D  i^i  B )  /\  -.  C  e.  ( C  i^i  B ) )  ->  ( D  i^i  B )  =/=  ( C  i^i  B ) )
2416, 22, 23syl2anc 665 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  C  e.  D )  ->  ( D  i^i  B
)  =/=  ( C  i^i  B ) )
2524necomd 2702 . . . . . 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 462 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  C )
27 simplrr 769 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  B )
2826, 27elind 3656 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  D  e.  ( C  i^i  B ) )
2910adantr 466 . . . . . . . . 9  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  Ord  D )
30 ordirr 5460 . . . . . . . . 9  |-  ( Ord 
D  ->  -.  D  e.  D )
3129, 30syl 17 . . . . . . . 8  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  D
)
32 inss1 3688 . . . . . . . . 9  |-  ( D  i^i  B )  C_  D
3332sseli 3466 . . . . . . . 8  |-  ( D  e.  ( D  i^i  B )  ->  D  e.  D )
3431, 33nsyl 124 . . . . . . 7  |-  ( ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B ) )  /\  D  e.  C )  ->  -.  D  e.  ( D  i^i  B ) )
35 nelne1 2760 . . . . . . 7  |-  ( ( D  e.  ( C  i^i  B )  /\  -.  D  e.  ( D  i^i  B ) )  ->  ( C  i^i  B )  =/=  ( D  i^i  B ) )
3628, 34, 35syl2anc 665 . . . . . 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 792 . . . . 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 435 . . . 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 238 . . 3  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( C  =/=  D  ->  ( C  i^i  B )  =/=  ( D  i^i  B
) ) )
4039necon4d 2658 . 2  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  ->  C  =  D )
)
41 ineq1 3663 . 2  |-  ( C  =  D  ->  ( C  i^i  B )  =  ( D  i^i  B
) )
4240, 41impbid1 206 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625    i^i cin 3441    C_ wss 3442   Ord word 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445
This theorem is referenced by:  fin23lem23  8754
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