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Theorem gchen2 8248
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 733 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  B  ~<_  ~P A )
2 gchi 8246 . . . . . 6  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323expia 1153 . . . . 5  |-  ( ( A  e. GCH  /\  A  ~<  B )  ->  ( B  ~<  ~P A  ->  A  e.  Fin )
)
43con3and 428 . . . 4  |-  ( ( ( A  e. GCH  /\  A  ~<  B )  /\  -.  A  e.  Fin )  ->  -.  B  ~<  ~P A )
54an32s 779 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  A  ~<  B )  ->  -.  B  ~<  ~P A )
65adantrr 697 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<  B  /\  B  ~<_  ~P A
) )  ->  -.  B  ~<  ~P A )
7 bren2 6892 . 2  |-  ( B 
~~  ~P A  <->  ( B  ~<_  ~P A  /\  -.  B  ~<  ~P A ) )
81, 6, 7sylanbrc 645 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 358    e. wcel 1684   ~Pcpw 3625   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863  GCHcgch 8242
This theorem is referenced by:  gchhar  8293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-f1o 5262  df-en 6864  df-dom 6865  df-sdom 6866  df-gch 8243
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