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Theorem prnzg 4286
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 4270 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 ne0i 3902 . 2 (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
31, 2syl 17 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wne 2790  c0 3896  {cpr 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3191  df-dif 3562  df-un 3564  df-nul 3897  df-sn 4154  df-pr 4156
This theorem is referenced by:  0nelop  4925  fr2nr  5057  mreincl  16191  subrgin  18735  lssincl  18897  incld  20770  umgrnloopv  25913  upgr1elem  25919  usgrnloopvALT  26003  difelsiga  30001  inelpisys  30022  inidl  33496  pmapmeet  34574  diameetN  35860  dihmeetlem2N  36103  dihmeetcN  36106  dihmeet  36147
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