Theorem prmg 3789

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 3785	. 2 |- ( A e. V -> E. x x e. { A } )
2		orc 717	. . . 4 |- ( x = A -> ( x = A \/ x = B ) )
3		velsn 3683	. . . 4 |- ( x e. { A } <-> x = A )
4		vex 2802	. . . . 5 |- x e. _V
5	4	elpr 3687	. . . 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 1646	. 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 713   = wceq 1395   E.wex 1538   e. wcel 2200   {csn 3666   {cpr 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 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prm  3791  opm  4320  onintexmid  4665  subrngin  14177  subrgin  14208  lssincl  14349  wlkvtxiedg  16056  wlkvtxiedgg  16057