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Theorem prnz 4152
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 4141 . 2  |-  A  e. 
{ A ,  B }
3 ne0i 3796 . 2  |-  ( A  e.  { A ,  B }  ->  { A ,  B }  =/=  (/) )
42, 3ax-mp 5 1  |-  { A ,  B }  =/=  (/)
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767    =/= wne 2662   _Vcvv 3118   (/)c0 3790   {cpr 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-nul 3791  df-sn 4034  df-pr 4036
This theorem is referenced by:  prnzg  4153  opnz  4724  fiint  7809  wilthlem2  23209  edgwlk  24354  umgrabi  24806  shincli  26103  chincli  26201
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