Theorem prmg 3639

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 3636	. 2 |- ( A e. V -> E. x x e. { A } )
2		orc 701	. . . 4 |- ( x = A -> ( x = A \/ x = B ) )
3		velsn 3539	. . . 4 |- ( x e. { A } <-> x = A )
4		vex 2684	. . . . 5 |- x e. _V
5	4	elpr 3543	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6	2, 3, 5	3imtr4i 200	. . 3 |- ( x e. { A } -> x e. { A , B } )
7	6	eximi 1579	. 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 697   = wceq 1331   E.wex 1468   e. wcel 1480   {csn 3522   {cpr 3523

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  prm  3641  opm  4151  onintexmid  4482