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Theorem gchi 8425
Description: The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchi  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )

Proof of Theorem gchi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 relsdom 7045 . . . . . . 7  |-  Rel  ~<
21brrelexi 4851 . . . . . 6  |-  ( B 
~<  ~P A  ->  B  e.  _V )
32adantl 453 . . . . 5  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  B  e.  _V )
4 breq2 4150 . . . . . . 7  |-  ( x  =  B  ->  ( A  ~<  x  <->  A  ~<  B ) )
5 breq1 4149 . . . . . . 7  |-  ( x  =  B  ->  (
x  ~<  ~P A  <->  B  ~<  ~P A ) )
64, 5anbi12d 692 . . . . . 6  |-  ( x  =  B  ->  (
( A  ~<  x  /\  x  ~<  ~P A
)  <->  ( A  ~<  B  /\  B  ~<  ~P A
) ) )
76spcegv 2973 . . . . 5  |-  ( B  e.  _V  ->  (
( A  ~<  B  /\  B  ~<  ~P A )  ->  E. x ( A 
~<  x  /\  x  ~<  ~P A ) ) )
83, 7mpcom 34 . . . 4  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  E. x ( A 
~<  x  /\  x  ~<  ~P A ) )
9 df-ex 1548 . . . 4  |-  ( E. x ( A  ~<  x  /\  x  ~<  ~P A
)  <->  -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
108, 9sylib 189 . . 3  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
11 elgch 8423 . . . . . 6  |-  ( A  e. GCH  ->  ( A  e. GCH  <->  ( A  e.  Fin  \/  A. x  -.  ( A 
~<  x  /\  x  ~<  ~P A ) ) ) )
1211ibi 233 . . . . 5  |-  ( A  e. GCH  ->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
1312orcomd 378 . . . 4  |-  ( A  e. GCH  ->  ( A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
)  \/  A  e. 
Fin ) )
1413ord 367 . . 3  |-  ( A  e. GCH  ->  ( -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
)  ->  A  e.  Fin ) )
1510, 14syl5 30 . 2  |-  ( A  e. GCH  ->  ( ( A 
~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin ) )
16153impib 1151 1  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2892   ~Pcpw 3735   class class class wbr 4146    ~< csdm 7037   Fincfn 7038  GCHcgch 8421
This theorem is referenced by:  gchen1  8426  gchen2  8427  gchpwdom  8475  gchaleph  8476
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-dom 7040  df-sdom 7041  df-gch 8422
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