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Theorem gchen2 9016
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 9014 . . . . . 6  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323expia 1198 . . . . 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 7558 . 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 1767   ~Pcpw 4016   class class class wbr 4453    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527   Fincfn 7528  GCHcgch 9010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-f1o 5601  df-en 7529  df-dom 7530  df-sdom 7531  df-gch 9011
This theorem is referenced by:  gchhar  9069
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