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Theorem prnzg 4783
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 4765 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
21ne0d 4348 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2938  c0 4339  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  preqsnd  4864  0nelop  5506  fr2nr  5666  mreincl  17644  subrngin  20578  subrgin  20613  lssincl  20981  incld  23067  umgrnloopv  29138  upgr1elem  29144  usgrnloopvALT  29233  difelsiga  34114  inelpisys  34135  inidl  38017  coss0  38461  pmapmeet  39756  diameetN  41039  dihmeetlem2N  41282  dihmeetcN  41285  dihmeet  41326
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