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Theorem gchen2 8465
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 734 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 gchi 8463 . . . . . 6  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323expia 1155 . . . . 5  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
43con3and 429 . . . 4  |-  ( ( ( A  e. GCH  /\  A  ~<  B )  /\  -.  A  e.  Fin )  ->  -.  B  ~<  ~P A )
54an32s 780 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<  B )  ->  -.  B  ~<  ~P A )
65adantrr 698 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  -.  B  ~<  ~P A )
7 bren2 7105 . 2  |-  ( B 
~~  ~P A  <->  ( B  ~<_  ~P A  /\  -.  B  ~<  ~P A ) )
81, 6, 7sylanbrc 646 1  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  B  ~~  ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1721   ~Pcpw 3767   class class class wbr 4180    ~~ cen 7073    ~<_ cdom 7074    ~< csdm 7075   Fincfn 7076  GCHcgch 8459
This theorem is referenced by:  gchhar  8510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-f1o 5428  df-en 7077  df-dom 7078  df-sdom 7079  df-gch 8460
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