Theorem bj-sels 13112

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 3554	. . 3 |- ( A e. V -> A e. { A } )
2		bj-snexg 13110	. . . . 5 |- ( A e. V -> { A } e. _V )
3		sbcel2g 3023	. . . . 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 3030	. . . . . 6 |- ( { A } e. _V -> [_ { A } / x ]_ x = { A } )
6	2, 5	syl 14	. . . . 5 |- ( A e. V -> [_ { A } / x ]_ x = { A } )
7	6	eleq2d 2209	. . . 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 2995	1 |- ( A e. V -> E. x A e. x )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686   [.wsbc 2909   [_csb 3003   {csn 3527

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-pr 4131  ax-bdor 13014  ax-bdeq 13018  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by: (None)