Theorem bj-sels 13283

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 3561	. . 3 |- ( A e. V -> A e. { A } )
2		bj-snexg 13281	. . . . 5 |- ( A e. V -> { A } e. _V )
3		sbcel2g 3028	. . . . 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 3035	. . . . . 6 |- ( { A } e. _V -> [_ { A } / x ]_ x = { A } )
6	2, 5	syl 14	. . . . 5 |- ( A e. V -> [_ { A } / x ]_ x = { A } )
7	6	eleq2d 2210	. . . 4 |- ( A e. V -> ( A e. [_ { A } / x ]_ x <-> A e. { A } ) )
8	4, 7	bitrd 187	. . 3 |- ( A e. V -> ( [. { A } / x ]. A e. x <-> A e. { A } ) )
9	1, 8	mpbird 166	. 2 |- ( A e. V -> [. { A } / x ]. A e. x )
10	9	spesbcd 2999	1 |- ( A e. V -> E. x A e. x )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   [.wsbc 2913   [_csb 3007   {csn 3532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-pr 4139  ax-bdor 13185  ax-bdeq 13189  ax-bdsep 13253

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by: (None)