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Theorem gchi 8262
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 6886 . . . . . . 7  |-  Rel  ~<
21brrelexi 4745 . . . . . 6  |-  ( B 
~<  ~P A  ->  B  e.  _V )
32adantl 452 . . . . 5  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  B  e.  _V )
4 breq2 4043 . . . . . . 7  |-  ( x  =  B  ->  ( A  ~<  x  <->  A  ~<  B ) )
5 breq1 4042 . . . . . . 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 2882 . . . . 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 1532 . . . 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 8260 . . . . . 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 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   ~Pcpw 3638   class class class wbr 4039    ~< csdm 6878   Fincfn 6879  GCHcgch 8258
This theorem is referenced by:  gchen1  8263  gchen2  8264  gchpwdom  8312  gchaleph  8313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dom 6881  df-sdom 6882  df-gch 8259
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