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Theorem gchen2 8338
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 8336 . . . . . 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 6980 . 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 1710   ~Pcpw 3701   class class class wbr 4104    ~~ cen 6948    ~<_ cdom 6949    ~< csdm 6950   Fincfn 6951  GCHcgch 8332
This theorem is referenced by:  gchhar  8383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-f1o 5344  df-en 6952  df-dom 6953  df-sdom 6954  df-gch 8333
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