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Theorem gchen1 9068
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 772 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  A  ~<_  B )
2 gchi 9067 . . . . . . 7  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
323com23 1237 . . . . . 6  |-  ( ( A  e. GCH  /\  B  ~<  ~P A  /\  A  ~<  B )  ->  A  e.  Fin )
433expia 1233 . . . . 5  |-  ( ( A  e. GCH  /\  B  ~<  ~P A )  -> 
( A  ~<  B  ->  A  e.  Fin )
)
54con3dimp 448 . . . 4  |-  ( ( ( A  e. GCH  /\  B  ~<  ~P A )  /\  -.  A  e. 
Fin )  ->  -.  A  ~<  B )
65an32s 821 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  B  ~<  ~P A
)  ->  -.  A  ~<  B )
76adantrl 730 . 2  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A
) )  ->  -.  A  ~<  B )
8 bren2 7618 . 2  |-  ( A 
~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
91, 7, 8sylanbrc 677 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 376    e. wcel 1904   ~Pcpw 3942   class class class wbr 4395    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587  GCHcgch 9063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-f1o 5596  df-en 7588  df-dom 7589  df-sdom 7590  df-gch 9064
This theorem is referenced by:  gchor  9070  gchcda1  9099  gchcdaidm  9111  gchxpidm  9112  gchhar  9122
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