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Theorem gchen1 8490
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 8489 . . . . . . 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 7130 . 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 1725   ~Pcpw 3791   class class class wbr 4204    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   Fincfn 7101  GCHcgch 8485
This theorem is referenced by:  gchor  8492  gchcda1  8521  gchcdaidm  8533  gchxpidm  8534  gchhar  8536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-f1o 5453  df-en 7102  df-dom 7103  df-sdom 7104  df-gch 8486
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