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Theorem fingch 8462
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 3478 . 2  |-  Fin  C_  ( Fin  u.  { x  | 
A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
2 df-gch 8460 . 2  |- GCH  =  ( Fin  u.  { x  |  A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
31, 2sseqtr4i 3349 1  |-  Fin  C_ GCH
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   A.wal 1546   {cab 2398    u. cun 3286    C_ wss 3288   ~Pcpw 3767   class class class wbr 4180    ~< csdm 7075   Fincfn 7076  GCHcgch 8459
This theorem is referenced by:  gch2  8518
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-un 3293  df-in 3295  df-ss 3302  df-gch 8460
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