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Theorem prnz 3745
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1  |-  A  e. 
_V
Assertion
Ref Expression
prnz  |-  { A ,  B }  =/=  (/)

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3  |-  A  e. 
_V
21prid1 3734 . 2  |-  A  e. 
{ A ,  B }
3 ne0i 3461 . 2  |-  ( A  e.  { A ,  B }  ->  { A ,  B }  =/=  (/) )
42, 3ax-mp 8 1  |-  { A ,  B }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {cpr 3641
This theorem is referenced by:  prnzg  3746  opnz  4242  fiint  7133  wilthlem2  20307  shincli  21941  chincli  22039  umgrabi  23907  toplat  25290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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