Theorem bj-sels 10705

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 3423	. . 3 |- ( A e. V -> A e. { A } )
2		bj-snexg 10703	. . . . 5 |- ( A e. V -> { A } e. _V )
3		sbcel2g 2927	. . . . 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 2933	. . . . . 6 |- ( { A } e. _V -> [_ { A } / x ]_ x = { A } )
6	2, 5	syl 14	. . . . 5 |- ( A e. V -> [_ { A } / x ]_ x = { A } )
7	6	eleq2d 2148	. . . 4 |- ( A e. V -> ( A e. [_ { A } / x ]_ x <-> A e. { A } ) )
8	4, 7	bitrd 186	. . 3 |- ( A e. V -> ( [. { A } / x ]. A e. x <-> A e. { A } ) )
9	1, 8	mpbird 165	. 2 |- ( A e. V -> [. { A } / x ]. A e. x )
10	9	spesbcd 2900	1 |- ( A e. V -> E. x A e. x )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   [.wsbc 2815   [_csb 2908   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-pr 3964  ax-bdor 10607  ax-bdeq 10611  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)