Theorem snmg 3810

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

Assertion

Ref	Expression
snmg	|- ( A e. V -> E. x x e. { A } )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem snmg

Step	Hyp	Ref	Expression
1		snidg 3718	. 2 |- ( A e. V -> A e. { A } )
2		elex2 2830	. 2 |- ( A e. { A } -> E. x x e. { A } )
3	1, 2	syl 14	1 |- ( A e. V -> E. x x e. { A } )

Syntax hints:   -> wi 4   E.wex 1541   e. wcel 2203   {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:  snm  3812  prmg  3814  exmidsssnc  4316  xpimasn  5211  1stconst  6417  2ndconst  6418  pwsbas  13505  lsssn0  14518