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Theorem gchor 8994
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 755 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 brdom2 7538 . . 3  |-  ( B  ~<_  ~P A  <->  ( B  ~<  ~P A  \/  B  ~~  ~P A ) )
31, 2sylib 196 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  \/  B  ~~  ~P A ) )
4 gchen1 8992 . . . . 5  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )
54expr 613 . . . 4  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<_  B )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
65adantrr 714 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
76orim1d 837 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    e. wcel 1823   ~Pcpw 3999   class class class wbr 4439    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508   Fincfn 7509  GCHcgch 8987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-f1o 5577  df-en 7510  df-dom 7511  df-sdom 7512  df-gch 8988
This theorem is referenced by:  gchdomtri  8996  gchpwdom  9037
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