Theorem prmg 3739

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 3736	. 2 |- ( A e. V -> E. x x e. { A } )
2		orc 713	. . . 4 |- ( x = A -> ( x = A \/ x = B ) )
3		velsn 3635	. . . 4 |- ( x e. { A } <-> x = A )
4		vex 2763	. . . . 5 |- x e. _V
5	4	elpr 3639	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6	2, 3, 5	3imtr4i 201	. . 3 |- ( x e. { A } -> x e. { A , B } )
7	6	eximi 1611	. 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 709   = wceq 1364   E.wex 1503   e. wcel 2164   {csn 3618   {cpr 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625

This theorem is referenced by:  prm  3741  opm  4263  onintexmid  4605  subrngin  13709  subrgin  13740  lssincl  13881