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Theorem prnzg 3759
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 3719 . . 3  |-  ( x  =  A  ->  { x ,  B }  =  { A ,  B }
)
21neeq1d 2472 . 2  |-  ( x  =  A  ->  ( { x ,  B }  =/=  (/)  <->  { A ,  B }  =/=  (/) ) )
3 vex 2804 . . 3  |-  x  e. 
_V
43prnz 3758 . 2  |-  { x ,  B }  =/=  (/)
52, 4vtoclg 2856 1  |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {cpr 3654
This theorem is referenced by:  0nelop  4272  fr2nr  4387  mreincl  13517  subrgin  15584  lssincl  15738  incld  16796  difelsiga  23509  umgra1  23893  inttop4  25620  inidl  26758  uslgra1  28252  usgra1  28253  pmapmeet  30584  diameetN  31868  dihmeetlem2N  32111  dihmeetcN  32114  dihmeet  32155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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