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Theorem prnzg 4778
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 4760 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
21ne0d 4342 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wne 2940  c0 4333  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by:  preqsnd  4859  0nelop  5501  fr2nr  5662  mreincl  17642  subrngin  20561  subrgin  20596  lssincl  20963  incld  23051  umgrnloopv  29123  upgr1elem  29129  usgrnloopvALT  29218  difelsiga  34134  inelpisys  34155  inidl  38037  coss0  38480  pmapmeet  39775  diameetN  41058  dihmeetlem2N  41301  dihmeetcN  41304  dihmeet  41345
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