Theorem prnzg 3757

Description: A pair containing a set is not empty. It is also inhabited (see prmg 3754). (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 3710	. . 3 |- ( x = A -> { x , B } = { A , B } )
2	1	neeq1d 2394	. 2 |- ( x = A -> ( { x , B } =/= (/) <-> { A , B } =/= (/) ) )
3		vex 2775	. . 3 |- x e. _V
4	3	prnz 3755	. 2 |- { x , B } =/= (/)
5	2, 4	vtoclg 2833	1 |- ( A e. V -> { A , B } =/= (/) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   =/= wne 2376   (/)c0 3460   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-dif 3168  df-un 3170  df-nul 3461  df-sn 3639  df-pr 3640

This theorem is referenced by:  0nelop  4293