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Theorem prnz 3464
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 A V
Assertion
Ref Expression
prnz {A, B} ≠ ∅

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 A V
21prid1 3450 . 2 A {A, B}
3 ne0i 3207 . 2 (A {A, B} → {A, B} ≠ ∅)
42, 3ax-mp 7 1 {A, B} ≠ ∅
Colors of variables: wff set class
Syntax hints:   wcel 1374  wne 2186  Vcvv 2535  c0 3201  {cpr 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357
This theorem is referenced by:  prnzg  3466
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