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Theorem prnz 3838
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 3827 . 2  |-  A  e. 
{ A ,  B }
3 ne0i 3549 . 2  |-  ( A  e.  { A ,  B }  ->  { A ,  B }  =/=  (/) )
42, 3ax-mp 8 1  |-  { A ,  B }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1715    =/= wne 2529   _Vcvv 2873   (/)c0 3543   {cpr 3730
This theorem is referenced by:  prnzg  3839  opnz  4345  fiint  7280  wilthlem2  20530  shincli  22254  chincli  22352  umgrabi  24494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-v 2875  df-dif 3241  df-un 3243  df-nul 3544  df-sn 3735  df-pr 3736
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