Theorem prnzg 3801

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3798). (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 3752	. . 3 |- ( x = A -> { x , B } = { A , B } )
2	1	neeq1d 2421	. 2 |- ( x = A -> ( { x , B } =/= (/) <-> { A , B } =/= (/) ) )
3		vex 2806	. . 3 |- x e. _V
4	3	prnz 3799	. 2 |- { x , B } =/= (/)
5	2, 4	vtoclg 2865	1 |- ( A e. V -> { A , B } =/= (/) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   (/)c0 3496   {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-in1 619  ax-in2 620  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-ne 2404  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-sn 3679  df-pr 3680

This theorem is referenced by:  0nelop  4346