Theorem prmg 3511

Description: A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)

Assertion

Ref	Expression
prmg	|- ( A e. V -> E. x x e. { A , B } )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem prmg

Step	Hyp	Ref	Expression
1		snmg 3508	. 2 |- ( A e. V -> E. x x e. { A } )
2		orc 665	. . . 4 |- ( x = A -> ( x = A \/ x = B ) )
3		velsn 3415	. . . 4 |- ( x e. { A } <-> x = A )
4		vex 2604	. . . . 5 |- x e. _V
5	4	elpr 3419	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6	2, 3, 5	3imtr4i 199	. . 3 |- ( x e. { A } -> x e. { A , B } )
7	6	eximi 1531	. 2 |- ( E. x x e. { A } -> E. x x e. { A , B } )
8	1, 7	syl 14	1 |- ( A e. V -> E. x x e. { A , B } )

Syntax hints:   -> wi 4   \/ wo 661   = wceq 1284   E.wex 1421   e. wcel 1433   {csn 3398   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by:  prm  3513  opm  3989  onintexmid  4315