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Theorem prnz 3481
 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 3467 . 2 A {A, B}
3 ne0i 3224 . 2 (A {A, B} → {A, B} ≠ ∅)
42, 3ax-mp 7 1 {A, B} ≠ ∅
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1390   ≠ wne 2201  Vcvv 2551  ∅c0 3218  {cpr 3368 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374 This theorem is referenced by:  prnzg  3483
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