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Theorem prnzg 4254
 Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
prnzg (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Proof of Theorem prnzg
StepHypRef Expression
1 prid1g 4239 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 ne0i 3880 . 2 (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
31, 2syl 17 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  0nelop  4886  fr2nr  5016  mreincl  16082  subrgin  18626  lssincl  18786  incld  20657  umgrnloopv  25772  upgr1elem  25778  umgra1  25855  uslgra1  25901  usgranloopv  25907  difelsiga  29523  inelpisys  29544  inidl  32999  pmapmeet  34077  diameetN  35363  dihmeetlem2N  35606  dihmeetcN  35609  dihmeet  35650  usgrnloopvALT  40428
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