Theorem snmg 3788

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 3696	. 2 |- ( A e. V -> A e. { A } )
2		elex2 2817	. 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 1538   e. wcel 2200   {csn 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-sn 3673

This theorem is referenced by:  snm  3790  prmg  3792  exmidsssnc  4291  xpimasn  5183  1stconst  6381  2ndconst  6382  pwsbas  13365  lsssn0  14374