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Theorem prnzg 4092
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 4051 . . 3  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
21neeq1d 2680 . 2  |-  ( x  =  A  ->  ( { x ,  B }  =/=  (/)  <->  { A ,  B }  =/=  (/) ) )
3 vex 3062 . . 3  |-  x  e. 
_V
43prnz 4091 . 2  |-  { x ,  B }  =/=  (/)
52, 4vtoclg 3117 1  |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   (/)c0 3738   {cpr 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3061  df-dif 3417  df-un 3419  df-nul 3739  df-sn 3973  df-pr 3975
This theorem is referenced by:  0nelop  4680  fr2nr  4801  mreincl  15213  subrgin  17772  lssincl  17931  incld  19836  umgra1  24743  uslgra1  24789  usgranloopv  24795  difelsiga  28581  inelpisys  28602  inidl  31709  pmapmeet  32790  diameetN  34076  dihmeetlem2N  34319  dihmeetcN  34322  dihmeet  34363
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