Theorem bj-sels 16049

Description: If a class is a set, then it is a member of a set. (Copied from set.mm.) (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 3672	. . 3 |- ( A e. V -> A e. { A } )
2		bj-snexg 16047	. . . . 5 |- ( A e. V -> { A } e. _V )
3		sbcel2g 3122	. . . . 5 |- ( { A } e. _V -> ( [. { A } / x ]. A e. x <-> A e. [_ { A } / x ]_ x ) )
4	2, 3	syl 14	. . . 4 |- ( A e. V -> ( [. { A } / x ]. A e. x <-> A e. [_ { A } / x ]_ x ) )
5		csbvarg 3129	. . . . . 6 |- ( { A } e. _V -> [_ { A } / x ]_ x = { A } )
6	2, 5	syl 14	. . . . 5 |- ( A e. V -> [_ { A } / x ]_ x = { A } )
7	6	eleq2d 2277	. . . 4 |- ( A e. V -> ( A e. [_ { A } / x ]_ x <-> A e. { A } ) )
8	4, 7	bitrd 188	. . 3 |- ( A e. V -> ( [. { A } / x ]. A e. x <-> A e. { A } ) )
9	1, 8	mpbird 167	. 2 |- ( A e. V -> [. { A } / x ]. A e. x )
10	9	spesbcd 3093	1 |- ( A e. V -> E. x A e. x )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   [.wsbc 3005   [_csb 3101   {csn 3643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-pr 4269  ax-bdor 15951  ax-bdeq 15955  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by: (None)