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Theorem fin23lem24 8158
Description: Lemma for fin23 8225. 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 3309 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  C  e.  A )
5 ordelord 4563 . . . . . 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 3309 . . . . . 6  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  D  e.  A )
9 ordelord 4563 . . . . . 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 4577 . . . . . 6  |-  ( ( Ord  C  /\  Ord  D )  ->  ( C  =  D  <->  -.  ( C  e.  D  \/  D  e.  C ) ) )
1211necon2abid 2624 . . . . 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 3490 . . . . . . . . 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 4559 . . . . . . . . . 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 3521 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  C
2221sseli 3304 . . . . . . . . 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 2656 . . . . . . . 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 2650 . . . . . 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 3490 . . . . . . . 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 4559 . . . . . . . . 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 3521 . . . . . . . . 9  |-  ( D  i^i  B )  C_  D
3534sseli 3304 . . . . . . . 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 2656 . . . . . . 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 2630 . 2  |-  ( ( ( Ord  A  /\  B  C_  A )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( C  i^i  B
)  =  ( D  i^i  B )  ->  C  =  D )
)
43 ineq1 3495 . 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 1649    e. wcel 1721    =/= wne 2567    i^i cin 3279    C_ wss 3280   Ord word 4540
This theorem is referenced by:  fin23lem23  8162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544
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