Theorem prmg 3798

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 3794	. 2 |- ( A e. V -> E. x x e. { A } )
2		orc 720	. . . 4 |- ( x = A -> ( x = A \/ x = B ) )
3		velsn 3690	. . . 4 |- ( x e. { A } <-> x = A )
4		vex 2806	. . . . 5 |- x e. _V
5	4	elpr 3694	. . . 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 1649	. 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 716   = wceq 1398   E.wex 1541   e. wcel 2202   {csn 3673   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  prm  3800  opm  4332  onintexmid  4677  subrngin  14308  subrgin  14339  lssincl  14481  wlkvtxiedg  16286  wlkvtxiedgg  16287