Theorem isfin7 9123

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

Assertion

Ref	Expression
isfin7	|- ( A e. V -> ( A e. FinVII <-> -. E. y e. ( On \ om ) A ~~ y ) )

Distinct variable group:   y, A

Allowed substitution hint:   V( y)

Proof of Theorem isfin7

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . 4 |- ( x = A -> ( x ~~ y <-> A ~~ y ) )
2	1	rexbidv 3052	. . 3 |- ( x = A -> ( E. y e. ( On \ om ) x ~~ y <-> E. y e. ( On \ om ) A ~~ y ) )
3	2	notbid 308	. 2 |- ( x = A -> ( -. E. y e. ( On \ om ) x ~~ y <-> -. E. y e. ( On \ om ) A ~~ y ) )
4		df-fin7 9113	. 2 |- FinVII = { x | -. E. y e. ( On \ om ) x ~~ y }
5	3, 4	elab2g 3353	1 |- ( A e. V -> ( A e. FinVII <-> -. E. y e. ( On \ om ) A ~~ y ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   class class class wbr 4653   Oncon0 5723   omcom 7065   ~~ cen 7952  Fin^VIIcfin7 9106

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-3an 1039  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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-fin7 9113

This theorem is referenced by:  fin17  9216  fin67  9217  isfin7-2  9218