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Theorem fingch 8503
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 3512 . 2  |-  Fin  C_  ( Fin  u.  { x  | 
A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
2 df-gch 8501 . 2  |- GCH  =  ( Fin  u.  { x  |  A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
31, 2sseqtr4i 3383 1  |-  Fin  C_ GCH
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360   A.wal 1550   {cab 2424    u. cun 3320    C_ wss 3322   ~Pcpw 3801   class class class wbr 4215    ~< csdm 7111   Fincfn 7112  GCHcgch 8500
This theorem is referenced by:  gch2  8555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-in 3329  df-ss 3336  df-gch 8501
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