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Theorem gchen2 8793
Description: If  A  <  B  <_  ~P A, and  A is an infinite GCH-set, then  B  =  ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchen2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  B  ~~  ~P A )

Proof of Theorem gchen2
StepHypRef Expression
1 simprr 756 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 gchi 8791 . . . . . 6  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323expia 1189 . . . . 5  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
43con3dimp 441 . . . 4  |-  ( ( ( A  e. GCH  /\  A  ~<  B )  /\  -.  A  e.  Fin )  ->  -.  B  ~<  ~P A )
54an32s 802 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<  B )  ->  -.  B  ~<  ~P A )
65adantrr 716 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  -.  B  ~<  ~P A )
7 bren2 7340 . 2  |-  ( B 
~~  ~P A  <->  ( B  ~<_  ~P A  /\  -.  B  ~<  ~P A ) )
81, 6, 7sylanbrc 664 1  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  B  ~~  ~P A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1756   ~Pcpw 3860   class class class wbr 4292    ~~ cen 7307    ~<_ cdom 7308    ~< csdm 7309   Fincfn 7310  GCHcgch 8787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-f1o 5425  df-en 7311  df-dom 7312  df-sdom 7313  df-gch 8788
This theorem is referenced by:  gchhar  8846
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