Theorem bj-sels 14548

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 3621	. . 3 |- ( A e. V -> A e. { A } )
2		bj-snexg 14546	. . . . 5 |- ( A e. V -> { A } e. _V )
3		sbcel2g 3078	. . . . 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 3085	. . . . . 6 |- ( { A } e. _V -> [_ { A } / x ]_ x = { A } )
6	2, 5	syl 14	. . . . 5 |- ( A e. V -> [_ { A } / x ]_ x = { A } )
7	6	eleq2d 2247	. . . 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 3049	1 |- ( A e. V -> E. x A e. x )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   [.wsbc 2962   [_csb 3057   {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-pr 4209  ax-bdor 14450  ax-bdeq 14454  ax-bdsep 14518

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3598  df-pr 3599

This theorem is referenced by: (None)