Theorem bj-sels 32950

Description: If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)

Assertion

Ref	Expression
bj-sels	|- ( A e. V -> E. x A e. x )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem bj-sels

Step	Hyp	Ref	Expression
1		snidg 4206	. . 3 |- ( A e. V -> A e. { A } )
2		sbcel2 3989	. . . 4 |- ( [. { A } / x ]. A e. x <-> A e. [_ { A } / x ]_ x )
3		snex 4908	. . . . . 6 |- { A } e. _V
4		csbvarg 4003	. . . . . 6 |- ( { A } e. _V -> [_ { A } / x ]_ x = { A } )
5	3, 4	ax-mp 5	. . . . 5 |- [_ { A } / x ]_ x = { A }
6	5	eleq2i 2693	. . . 4 |- ( A e. [_ { A } / x ]_ x <-> A e. { A } )
7	2, 6	bitri 264	. . 3 |- ( [. { A } / x ]. A e. x <-> A e. { A } )
8	1, 7	sylibr 224	. 2 |- ( A e. V -> [. { A } / x ]. A e. x )
9	8	spesbcd 3522	1 |- ( A e. V -> E. x A e. x )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   {csn 4177

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  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-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)