Theorem snmg 3649

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 3561	. 2 |- ( A e. V -> A e. { A } )
2		elex2 2705	. 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 1469   e. wcel 1481   {csn 3532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538

This theorem is referenced by:  snm  3651  prmg  3652  exmidsssnc  4134  xpimasn  4995  1stconst  6126  2ndconst  6127