Theorem prnzg 3746

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3743). (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
prnzg	|- ( A e. V -> { A , B } =/= (/) )

Proof of Theorem prnzg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 3699	. . 3 |- ( x = A -> { x , B } = { A , B } )
2	1	neeq1d 2385	. 2 |- ( x = A -> ( { x , B } =/= (/) <-> { A , B } =/= (/) ) )
3		vex 2766	. . 3 |- x e. _V
4	3	prnz 3744	. 2 |- { x , B } =/= (/)
5	2, 4	vtoclg 2824	1 |- ( A e. V -> { A , B } =/= (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   =/= wne 2367   (/)c0 3450   {cpr 3623

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-un 3161  df-nul 3451  df-sn 3628  df-pr 3629

This theorem is referenced by:  0nelop  4281