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	⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Proof of Theorem prnzg

Step	Hyp	Ref	Expression
1		prid1g 4765	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2	1	ne0d 4336	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2107   ≠ wne 2941  ∅c0 4323  {cpr 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324  df-sn 4630  df-pr 4632

This theorem is referenced by:  preqsnd  4860  0nelop  5497  fr2nr  5655  mreincl  17543  subrgin  20343  lssincl  20576  incld  22547  umgrnloopv  28366  upgr1elem  28372  usgrnloopvALT  28458  difelsiga  33131  inelpisys  33152  inidl  36898  coss0  37349  pmapmeet  38644  diameetN  39927  dihmeetlem2N  40170  dihmeetcN  40173  dihmeet  40214  subrngin  46740