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Theorem prnzg 3924
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.
StepHypRef Expression
1 preq1 3883 . . 3  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
21neeq1d 2614 . 2  |-  ( x  =  A  ->  ( { x ,  B }  =/=  (/)  <->  { A ,  B }  =/=  (/) ) )
3 vex 2959 . . 3  |-  x  e. 
_V
43prnz 3923 . 2  |-  { x ,  B }  =/=  (/)
52, 4vtoclg 3011 1  |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   {cpr 3815
This theorem is referenced by:  0nelop  4446  fr2nr  4560  mreincl  13824  subrgin  15891  lssincl  16041  incld  17107  umgra1  21361  uslgra1  21392  usgranloopv  21398  difelsiga  24516  inidl  26640  pmapmeet  30570  diameetN  31854  dihmeetlem2N  32097  dihmeetcN  32100  dihmeet  32141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-sn 3820  df-pr 3821
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