Theorem prnzg 3716

Description: A pair containing a set is not empty. (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 3669	. . 3 |- ( x = A -> { x , B } = { A , B } )
2	1	neeq1d 2365	. 2 |- ( x = A -> ( { x , B } =/= (/) <-> { A , B } =/= (/) ) )
3		vex 2740	. . 3 |- x e. _V
4	3	prnz 3714	. 2 |- { x , B } =/= (/)
5	2, 4	vtoclg 2797	1 |- ( A e. V -> { A , B } =/= (/) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   =/= wne 2347   (/)c0 3422   {cpr 3593

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-dif 3131  df-un 3133  df-nul 3423  df-sn 3598  df-pr 3599

This theorem is referenced by:  0nelop  4247