Theorem snm 3812

Description: The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)

Hypothesis

Ref	Expression
snnz.1	|- A e. _V

Assertion

Ref	Expression
snm	|- E. x x e. { A }

Distinct variable group:   x, A

Proof of Theorem snm

Step	Hyp	Ref	Expression
1		snnz.1	. 2 |- A e. _V
2		snmg 3810	. 2 |- ( A e. _V -> E. x x e. { A } )
3	1, 2	ax-mp 5	1 |- E. x x e. { A }

Syntax hints:   E.wex 1541   e. wcel 2203   _Vcvv 2813   {csn 3689

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sn 3695

This theorem is referenced by:  mss  4342  ssfilem  7130  ssfilemd  7132  diffitest  7144  djuexb  7335  exmidonfinlem  7496  exmidfodomrlemim  7504  cc2lem  7580