Theorem isfin4 9119

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

Assertion

Ref	Expression
isfin4	|- ( A e. V -> ( A e. FinIV <-> -. E. y ( y C. A /\ y ~~ A ) ) )

Distinct variable group:   y, A

Allowed substitution hint:   V( y)

Proof of Theorem isfin4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psseq2 3695	. . . . 5 |- ( x = A -> ( y C. x <-> y C. A ) )
2		breq2 4657	. . . . 5 |- ( x = A -> ( y ~~ x <-> y ~~ A ) )
3	1, 2	anbi12d 747	. . . 4 |- ( x = A -> ( ( y C. x /\ y ~~ x ) <-> ( y C. A /\ y ~~ A ) ) )
4	3	exbidv 1850	. . 3 |- ( x = A -> ( E. y ( y C. x /\ y ~~ x ) <-> E. y ( y C. A /\ y ~~ A ) ) )
5	4	notbid 308	. 2 |- ( x = A -> ( -. E. y ( y C. x /\ y ~~ x ) <-> -. E. y ( y C. A /\ y ~~ A ) ) )
6		df-fin4 9109	. 2 |- FinIV = { x | -. E. y ( y C. x /\ y ~~ x ) }
7	5, 6	elab2g 3353	1 |- ( A e. V -> ( A e. FinIV <-> -. E. y ( y C. A /\ y ~~ A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   C. wpss 3575   class class class wbr 4653   ~~ cen 7952  Fin^IVcfin4 9102

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-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-fin4 9109

This theorem is referenced by:  fin4i  9120  fin4en1  9131  ssfin4  9132  infpssALT  9135  isfin4-2  9136