Theorem prmg 3704

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 3701	. 2 |- ( A e. V -> E. x x e. { A } )
2		orc 707	. . . 4 |- ( x = A -> ( x = A \/ x = B ) )
3		velsn 3600	. . . 4 |- ( x e. { A } <-> x = A )
4		vex 2733	. . . . 5 |- x e. _V
5	4	elpr 3604	. . . 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 1593	. 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 703   = wceq 1348   E.wex 1485   e. wcel 2141   {csn 3583   {cpr 3584

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by:  prm  3706  opm  4219  onintexmid  4557