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Theorem gchen1 8434
Description: If  A  <_  B  <  ~P A, and  A is an infinite GCH-set, then  A  =  B in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchen1  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )

Proof of Theorem gchen1
StepHypRef Expression
1 simprl 733 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~<_  B )
2 gchi 8433 . . . . . . 7  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323com23 1159 . . . . . 6  |-  ( ( A  e. GCH  /\  B  ~<  ~P A  /\  A  ~<  B )  ->  A  e.  Fin )
433expia 1155 . . . . 5  |-  ( ( A  e. GCH  /\  B  ~<  ~P A )  -> 
( A  ~<  B  ->  A  e.  Fin )
)
54con3and 429 . . . 4  |-  ( ( ( A  e. GCH  /\  B  ~<  ~P A )  /\  -.  A  e. 
Fin )  ->  -.  A  ~<  B )
65an32s 780 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  B  ~<  ~P A
)  ->  -.  A  ~<  B )
76adantrl 697 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  -.  A  ~<  B )
8 bren2 7075 . 2  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
91, 7, 8sylanbrc 646 1  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1717   ~Pcpw 3743   class class class wbr 4154    ~~ cen 7043    ~<_ cdom 7044    ~< csdm 7045   Fincfn 7046  GCHcgch 8429
This theorem is referenced by:  gchor  8436  gchcda1  8465  gchcdaidm  8477  gchxpidm  8478  gchhar  8480
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-f1o 5402  df-en 7047  df-dom 7048  df-sdom 7049  df-gch 8430
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