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Theorem gchen1 8263
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 732 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~<_  B )
2 gchi 8262 . . . . . . 7  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323com23 1157 . . . . . 6  |-  ( ( A  e. GCH  /\  B  ~<  ~P A  /\  A  ~<  B )  ->  A  e.  Fin )
433expia 1153 . . . . 5  |-  ( ( A  e. GCH  /\  B  ~<  ~P A )  -> 
( A  ~<  B  ->  A  e.  Fin )
)
54con3and 428 . . . 4  |-  ( ( ( A  e. GCH  /\  B  ~<  ~P A )  /\  -.  A  e. 
Fin )  ->  -.  A  ~<  B )
65an32s 779 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  B  ~<  ~P A
)  ->  -.  A  ~<  B )
76adantrl 696 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  -.  A  ~<  B )
8 bren2 6908 . 2  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
91, 7, 8sylanbrc 645 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 358    e. wcel 1696   ~Pcpw 3638   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879  GCHcgch 8258
This theorem is referenced by:  gchor  8265  gchcda1  8294  gchcdaidm  8306  gchxpidm  8307  gchhar  8309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-f1o 5278  df-en 6880  df-dom 6881  df-sdom 6882  df-gch 8259
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