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Theorem gchor 8898
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 756 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 brdom2 7442 . . 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 8896 . . . . 5  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )
54expr 615 . . . 4  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<_  B )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
65adantrr 716 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A
) )  ->  ( B  ~<  ~P A  ->  A  ~~  B ) )
76orim1d 835 . 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 368    /\ wa 369    e. wcel 1758   ~Pcpw 3961   class class class wbr 4393    ~~ cen 7410    ~<_ cdom 7411    ~< csdm 7412   Fincfn 7413  GCHcgch 8891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-f1o 5526  df-en 7414  df-dom 7415  df-sdom 7416  df-gch 8892
This theorem is referenced by:  gchdomtri  8900  gchpwdom  8941
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