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Theorem gchen1 8811
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 755 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~<_  B )
2 gchi 8810 . . . . . . 7  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323com23 1193 . . . . . 6  |-  ( ( A  e. GCH  /\  B  ~<  ~P A  /\  A  ~<  B )  ->  A  e.  Fin )
433expia 1189 . . . . 5  |-  ( ( A  e. GCH  /\  B  ~<  ~P A )  -> 
( A  ~<  B  ->  A  e.  Fin )
)
54con3dimp 441 . . . 4  |-  ( ( ( A  e. GCH  /\  B  ~<  ~P A )  /\  -.  A  e. 
Fin )  ->  -.  A  ~<  B )
65an32s 802 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  B  ~<  ~P A
)  ->  -.  A  ~<  B )
76adantrl 715 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  -.  A  ~<  B )
8 bren2 7359 . 2  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
91, 7, 8sylanbrc 664 1  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1756   ~Pcpw 3879   class class class wbr 4311    ~~ cen 7326    ~<_ cdom 7327    ~< csdm 7328   Fincfn 7329  GCHcgch 8806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-br 4312  df-opab 4370  df-xp 4865  df-rel 4866  df-f1o 5444  df-en 7330  df-dom 7331  df-sdom 7332  df-gch 8807
This theorem is referenced by:  gchor  8813  gchcda1  8842  gchcdaidm  8854  gchxpidm  8855  gchhar  8865
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