Theorem snmb 3755

Description: A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)

Assertion

Ref	Expression
snmb	|- ( A e. _V <-> E. x x e. { A } )

Distinct variable group:   x, A

Proof of Theorem snmb

Step	Hyp	Ref	Expression
1		isset 2779	. 2 |- ( A e. _V <-> E. x x = A )
2		velsn 3651	. . 3 |- ( x e. { A } <-> x = A )
3	2	exbii 1629	. 2 |- ( E. x x e. { A } <-> E. x x = A )
4	1, 3	bitr4i 187	1 |- ( A e. _V <-> E. x x e. { A } )

Syntax hints:   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   {csn 3634

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-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sn 3640

This theorem is referenced by:  lpvtx  15719