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Theorem fingch 8245
Description: A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
fingch  |-  Fin  C_ GCH

Proof of Theorem fingch
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun1 3338 . 2  |-  Fin  C_  ( Fin  u.  { x  | 
A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
2 df-gch 8243 . 2  |- GCH  =  ( Fin  u.  { x  |  A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
31, 2sseqtr4i 3211 1  |-  Fin  C_ GCH
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   A.wal 1527   {cab 2269    u. cun 3150    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023    ~< csdm 6862   Fincfn 6863  GCHcgch 8242
This theorem is referenced by:  gch2  8301
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-gch 8243
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