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

Theorem fingch 8261
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 3351 . 2  |-  Fin  C_  ( Fin  u.  { x  | 
A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
2 df-gch 8259 . 2  |- GCH  =  ( Fin  u.  { x  |  A. y  -.  (
x  ~<  y  /\  y  ~<  ~P x ) } )
31, 2sseqtr4i 3224 1  |-  Fin  C_ GCH
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   A.wal 1530   {cab 2282    u. cun 3163    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039    ~< csdm 6878   Fincfn 6879  GCHcgch 8258
This theorem is referenced by:  gch2  8317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-gch 8259
  Copyright terms: Public domain W3C validator