Theorem prnzg 3792

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3789). (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 3743	. . 3 |- ( x = A -> { x , B } = { A , B } )
2	1	neeq1d 2418	. 2 |- ( x = A -> ( { x , B } =/= (/) <-> { A , B } =/= (/) ) )
3		vex 2802	. . 3 |- x e. _V
4	3	prnz 3790	. 2 |- { x , B } =/= (/)
5	2, 4	vtoclg 2861	1 |- ( A e. V -> { A , B } =/= (/) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   {cpr 3667

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-pr 3673

This theorem is referenced by:  0nelop  4334