MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gchi Structured version   Unicode version

Theorem gchi 9005
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 7525 . . . . . . 7  |-  Rel  ~<
21brrelexi 5030 . . . . . 6  |-  ( B 
~<  ~P A  ->  B  e.  _V )
32adantl 466 . . . . 5  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  B  e.  _V )
4 breq2 4441 . . . . . . 7  |-  ( x  =  B  ->  ( A  ~<  x  <->  A  ~<  B ) )
5 breq1 4440 . . . . . . 7  |-  ( x  =  B  ->  (
x  ~<  ~P A  <->  B  ~<  ~P A ) )
64, 5anbi12d 710 . . . . . 6  |-  ( x  =  B  ->  (
( A  ~<  x  /\  x  ~<  ~P A
)  <->  ( A  ~<  B  /\  B  ~<  ~P A
) ) )
76spcegv 3181 . . . . 5  |-  ( B  e.  _V  ->  (
( A  ~<  B  /\  B  ~<  ~P A )  ->  E. x ( A 
~<  x  /\  x  ~<  ~P A ) ) )
83, 7mpcom 36 . . . 4  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  E. x ( A 
~<  x  /\  x  ~<  ~P A ) )
9 df-ex 1600 . . . 4  |-  ( E. x ( A  ~<  x  /\  x  ~<  ~P A
)  <->  -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
108, 9sylib 196 . . 3  |-  ( ( A  ~<  B  /\  B  ~<  ~P A )  ->  -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) )
11 elgch 9003 . . . . . 6  |-  ( A  e. GCH  ->  ( A  e. GCH  <->  ( A  e.  Fin  \/  A. x  -.  ( A 
~<  x  /\  x  ~<  ~P A ) ) ) )
1211ibi 241 . . . . 5  |-  ( A  e. GCH  ->  ( A  e. 
Fin  \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
) ) )
1312orcomd 388 . . . 4  |-  ( A  e. GCH  ->  ( A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
)  \/  A  e. 
Fin ) )
1413ord 377 . . 3  |-  ( A  e. GCH  ->  ( -.  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A
)  ->  A  e.  Fin ) )
1510, 14syl5 32 . 2  |-  ( A  e. GCH  ->  ( ( A 
~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin ) )
16153impib 1195 1  |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 974   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095   ~Pcpw 3997   class class class wbr 4437    ~< csdm 7517   Fincfn 7518  GCHcgch 9001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-dom 7520  df-sdom 7521  df-gch 9002
This theorem is referenced by:  gchen1  9006  gchen2  9007  gchpwdom  9051  gchaleph  9052
  Copyright terms: Public domain W3C validator