Theorem isfin1a 8693

Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin1a

Distinct variable group:   ,

Proof of Theorem isfin1a

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4015	. . 3
2		difeq1 3614	. . . . 5
3	2	eleq1d 2526	. . . 4
4	3	orbi2d 701	. . 3
5	1, 4	raleqbidv 3068	. 2
6		df-fin1a 8686	. 2
7	5, 6	elab2g 3248	1

Syntax hints:  ->wi 4  <->wb 184  \/wo 368  =wceq 1395  e.wcel 1818  A.wral 2807  \cdif 3472  ~Pcpw 4012  cfn 7536  cfin1a 8679

This theorem is referenced by:  fin1ai  8694  fin11a  8784  enfin1ai  8785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-pw 4014  df-fin1a 8686