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Theorem prnz 3651
 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 3635 . 2 𝐴 ∈ {𝐴, 𝐵}
3 ne0i 3372 . 2 (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
42, 3ax-mp 5 1 {𝐴, 𝐵} ≠ ∅
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1481   ≠ wne 2309  Vcvv 2689  ∅c0 3366  {cpr 3531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3076  df-un 3078  df-nul 3367  df-sn 3536  df-pr 3537 This theorem is referenced by:  prnzg  3653
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