Theorem snmb 3767

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 2786	. 2 |- ( A e. _V <-> E. x x = A )
2		velsn 3663	. . 3 |- ( x e. { A } <-> x = A )
3	2	exbii 1631	. 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 1375   E.wex 1518   e. wcel 2180   _Vcvv 2779   {csn 3646

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-sn 3652

This theorem is referenced by:  lpvtx  15844