Theorem fingch 9445

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.

Step	Hyp	Ref	Expression
1		ssun1 3776	. 2 |- Fin C_ ( Fin u. { x | A. y -. ( x ~< y /\ y ~< ~P x ) } )
2		df-gch 9443	. 2 |- GCH = ( Fin u. { x | A. y -. ( x ~< y /\ y ~< ~P x ) } )
3	1, 2	sseqtr4i 3638	1 |- Fin C_ GCH

Syntax hints:   -. wn 3   /\ wa 384   A.wal 1481   {cab 2608   u. cun 3572   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ~< csdm 7954   Fincfn 7955  GCHcgch 9442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-gch 9443

This theorem is referenced by:  gch2  9497