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Theorem prnz 3592
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 3576 . 2 𝐴 ∈ {𝐴, 𝐵}
3 ne0i 3316 . 2 (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
42, 3ax-mp 7 1 {𝐴, 𝐵} ≠ ∅
Colors of variables: wff set class
Syntax hints:  wcel 1448  wne 2267  Vcvv 2641  c0 3310  {cpr 3475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-v 2643  df-dif 3023  df-un 3025  df-nul 3311  df-sn 3480  df-pr 3481
This theorem is referenced by:  prnzg  3594
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