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Theorem prnz 3545
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 3531 . 2  |-  A  e. 
{ A ,  B }
3 ne0i 3281 . 2  |-  ( A  e.  { A ,  B }  ->  { A ,  B }  =/=  (/) )
42, 3ax-mp 7 1  |-  { A ,  B }  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1436    =/= wne 2251   _Vcvv 2615   (/)c0 3275   {cpr 3432
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-v 2617  df-dif 2990  df-un 2992  df-nul 3276  df-sn 3437  df-pr 3438
This theorem is referenced by:  prnzg  3547
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