Theorem snmb 3797

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 2810	. 2 |- ( A e. _V <-> E. x x = A )
2		velsn 3690	. . 3 |- ( x e. { A } <-> x = A )
3	2	exbii 1654	. 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 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by:  lpvtx  16003