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Theorem gchor 8294
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 733 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 brdom2 6934 . . 3  |-  ( B  ~<_  ~P A  <->  ( B  ~<  ~P A  \/  B  ~~  ~P A ) )
31, 2sylib 188 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  \/  B  ~~  ~P A ) )
4 gchen1 8292 . . . . 5  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )
54expr 598 . . . 4  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<_  B )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
65adantrr 697 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
76orim1d 812 . 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 14 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 357    /\ wa 358    e. wcel 1701   ~Pcpw 3659   class class class wbr 4060    ~~ cen 6903    ~<_ cdom 6904    ~< csdm 6905   Fincfn 6906  GCHcgch 8287
This theorem is referenced by:  gchdomtri  8296  gchpwdom  8341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-f1o 5299  df-en 6907  df-dom 6908  df-sdom 6909  df-gch 8288
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