Theorem prnzg 4150

Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
prnzg

Proof of Theorem prnzg

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 4109	. . 3
2	1	neeq1d 2734	. 2
3		vex 3112	. . 3
4	3	prnz 4149	. 2
5	2, 4	vtoclg 3167	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  =/=wne 2652  c0 3784  {cpr 4031

This theorem is referenced by:  0nelop  4742  fr2nr  4862  mreincl  14996  subrgin  17452  lssincl  17611  incld  19544  umgra1  24326  uslgra1  24372  usgranloopv  24378  difelsiga  28133  inidl  30427  pmapmeet  35497  diameetN  36783  dihmeetlem2N  37026  dihmeetcN  37029  dihmeet  37070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-pr 4032