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Theorem gchor 8502
Description: If  A  <_  B  <_  ~P A, and  A is an infinite GCH-set, then either  A  =  B or  B  =  ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchor  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( A  ~~  B  \/  B  ~~  ~P A ) )

Proof of Theorem gchor
StepHypRef Expression
1 simprr 734 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 brdom2 7137 . . 3  |-  ( B  ~<_  ~P A  <->  ( B  ~<  ~P A  \/  B  ~~  ~P A ) )
31, 2sylib 189 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  \/  B  ~~  ~P A ) )
4 gchen1 8500 . . . . 5  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )
54expr 599 . . . 4  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<_  B )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
65adantrr 698 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
76orim1d 813 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  (
( B  ~<  ~P A  \/  B  ~~  ~P A
)  ->  ( A  ~~  B  \/  B  ~~  ~P A ) ) )
83, 7mpd 15 1  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( A  ~~  B  \/  B  ~~  ~P A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    e. wcel 1725   ~Pcpw 3799   class class class wbr 4212    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108   Fincfn 7109  GCHcgch 8495
This theorem is referenced by:  gchdomtri  8504  gchpwdom  8549
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-f1o 5461  df-en 7110  df-dom 7111  df-sdom 7112  df-gch 8496
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