Theorem abnex 6965

Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 6966 and pwnex 6968. See the comment of abnexg 6964. (Contributed by BJ, 2-May-2021.)

Assertion

Ref	Expression
abnex	|- ( A. x ( F e. V /\ x e. F ) -> -. { y | E. x y = F } e. _V )

Distinct variable groups:   x, y   y, F

Allowed substitution hints:   F( x)   V( x, y)

Proof of Theorem abnex

Step	Hyp	Ref	Expression
1		vprc 4796	. 2 |- -. _V e. _V
2		alral 2928	. . 3 |- ( A. x ( F e. V /\ x e. F ) -> A. x e. _V ( F e. V /\ x e. F ) )
3		rexv 3220	. . . . . . 7 |- ( E. x e. _V y = F <-> E. x y = F )
4	3	bicomi 214	. . . . . 6 |- ( E. x y = F <-> E. x e. _V y = F )
5	4	abbii 2739	. . . . 5 |- { y | E. x y = F } = { y | E. x e. _V y = F }
6	5	eleq1i 2692	. . . 4 |- ( { y | E. x y = F } e. _V <-> { y | E. x e. _V y = F } e. _V )
7	6	biimpi 206	. . 3 |- ( { y | E. x y = F } e. _V -> { y | E. x e. _V y = F } e. _V )
8		abnexg 6964	. . 3 |- ( A. x e. _V ( F e. V /\ x e. F ) -> ( { y | E. x e. _V y = F } e. _V -> _V e. _V ) )
9	2, 7, 8	syl2im 40	. 2 |- ( A. x ( F e. V /\ x e. F ) -> ( { y | E. x y = F } e. _V -> _V e. _V ) )
10	1, 9	mtoi 190	1 |- ( A. x ( F e. V /\ x e. F ) -> -. { y | E. x y = F } e. _V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-un 6949

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-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-sn 4178  df-uni 4437  df-iun 4522

This theorem is referenced by:  snnex  6966  pwnex  6968