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Theorem gchi 8455
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 7075 . . . . . . 7  |-  Rel  ~<
21brrelexi 4877 . . . . . 6  |-  ( B 
~<  ~P A  ->  B  e.  _V )
32adantl 453 . . . . 5  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  B  e.  _V )
4 breq2 4176 . . . . . . 7  |-  ( x  =  B  ->  ( A  ~<  x  <->  A  ~<  B ) )
5 breq1 4175 . . . . . . 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 2997 . . . . 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 8453 . . . . . 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 1721   _Vcvv 2916   ~Pcpw 3759   class class class wbr 4172    ~< csdm 7067   Fincfn 7068  GCHcgch 8451
This theorem is referenced by:  gchen1  8456  gchen2  8457  gchpwdom  8505  gchaleph  8506
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-dom 7070  df-sdom 7071  df-gch 8452
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