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Theorem gchi 8246
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 6870 . . . . . . 7  |-  Rel  ~<
21brrelexi 4729 . . . . . 6  |-  ( B 
~<  ~P A  ->  B  e.  _V )
32adantl 452 . . . . 5  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  B  e.  _V )
4 breq2 4027 . . . . . . 7  |-  ( x  =  B  ->  ( A  ~<  x  <->  A  ~<  B ) )
5 breq1 4026 . . . . . . 7  |-  ( x  =  B  ->  (
x  ~<  ~P A  <->  B  ~<  ~P A ) )
64, 5anbi12d 691 . . . . . 6  |-  ( x  =  B  ->  (
( A  ~<  x  /\  x  ~<  ~P A
)  <->  ( A  ~<  B  /\  B  ~<  ~P A
) ) )
76spcegv 2869 . . . . 5  |-  ( B  e.  _V  ->  (
( A  ~<  B  /\  B  ~<  ~P A )  ->  E. x ( A 
~<  x  /\  x  ~<  ~P A ) ) )
83, 7mpcom 32 . . . 4  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  E. x ( A 
~<  x  /\  x  ~<  ~P A ) )
9 df-ex 1529 . . . 4  |-  ( E. x ( A  ~<  x  /\  x  ~<  ~P A
)  <->  -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
108, 9sylib 188 . . 3  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
11 elgch 8244 . . . . . 6  |-  ( A  e. GCH  ->  ( A  e. GCH  <->  ( A  e.  Fin  \/  A. x  -.  ( A 
~<  x  /\  x  ~<  ~P A ) ) ) )
1211ibi 232 . . . . 5  |-  ( A  e. GCH  ->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
1312orcomd 377 . . . 4  |-  ( A  e. GCH  ->  ( A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
)  \/  A  e. 
Fin ) )
1413ord 366 . . 3  |-  ( A  e. GCH  ->  ( -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
)  ->  A  e.  Fin ) )
1510, 14syl5 28 . 2  |-  ( A  e. GCH  ->  ( ( A 
~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin ) )
16153impib 1149 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 357    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023    ~< csdm 6862   Fincfn 6863  GCHcgch 8242
This theorem is referenced by:  gchen1  8247  gchen2  8248  gchpwdom  8296  gchaleph  8297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dom 6865  df-sdom 6866  df-gch 8243
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