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Theorem fingch 8999
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 3649 . 2  |-  Fin  C_  ( Fin  u.  { x  | 
A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
2 df-gch 8997 . 2  |- GCH  =  ( Fin  u.  { x  |  A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
31, 2sseqtr4i 3519 1  |-  Fin  C_ GCH
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369   A.wal 1379   {cab 2426    u. cun 3456    C_ wss 3458   ~Pcpw 3993   class class class wbr 4433    ~< csdm 7513   Fincfn 7514  GCHcgch 8996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-un 3463  df-in 3465  df-ss 3472  df-gch 8997
This theorem is referenced by:  gch2  9051
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