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Theorem prnz 3536
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 3522 . 2  |-  A  e. 
{ A ,  B }
3 ne0i 3275 . 2  |-  ( A  e.  { A ,  B }  ->  { A ,  B }  =/=  (/) )
42, 3ax-mp 7 1  |-  { A ,  B }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1434    =/= wne 2249   _Vcvv 2612   (/)c0 3269   {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2614  df-dif 2986  df-un 2988  df-nul 3270  df-sn 3428  df-pr 3429
This theorem is referenced by:  prnzg  3538
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