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Theorem prnz 3883
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 3872 . 2  |-  A  e. 
{ A ,  B }
3 ne0i 3594 . 2  |-  ( A  e.  { A ,  B }  ->  { A ,  B }  =/=  (/) )
42, 3ax-mp 8 1  |-  { A ,  B }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   {cpr 3775
This theorem is referenced by:  prnzg  3884  opnz  4392  fiint  7342  wilthlem2  20805  umgrabi  21658  shincli  22817  chincli  22915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-nul 3589  df-sn 3780  df-pr 3781
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