Theorem isfin1a 9114

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

Assertion

Ref	Expression
isfin1a	|- ( A e. V -> ( A e. FinIa <-> A. y e. ~P A ( y e. Fin \/ ( A \ y ) e. Fin ) ) )

Distinct variable group:   y, A

Allowed substitution hint:   V( y)

Proof of Theorem isfin1a

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . 3 |- ( x = A -> ~P x = ~P A )
2		difeq1 3721	. . . . 5 |- ( x = A -> ( x \ y ) = ( A \ y ) )
3	2	eleq1d 2686	. . . 4 |- ( x = A -> ( ( x \ y ) e. Fin <-> ( A \ y ) e. Fin ) )
4	3	orbi2d 738	. . 3 |- ( x = A -> ( ( y e. Fin \/ ( x \ y ) e. Fin ) <-> ( y e. Fin \/ ( A \ y ) e. Fin ) ) )
5	1, 4	raleqbidv 3152	. 2 |- ( x = A -> ( A. y e. ~P x ( y e. Fin \/ ( x \ y ) e. Fin ) <-> A. y e. ~P A ( y e. Fin \/ ( A \ y ) e. Fin ) ) )
6		df-fin1a 9107	. 2 |- FinIa = { x | A. y e. ~P x ( y e. Fin \/ ( x \ y ) e. Fin ) }
7	5, 6	elab2g 3353	1 |- ( A e. V -> ( A e. FinIa <-> A. y e. ~P A ( y e. Fin \/ ( A \ y ) e. Fin ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   ~Pcpw 4158   Fincfn 7955  Fin^Iacfin1a 9100

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-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160  df-fin1a 9107

This theorem is referenced by:  fin1ai  9115  fin11a  9205  enfin1ai  9206