Theorem prnzg 4759

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 4741	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2	1	ne0d 4322	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2933  ∅c0 4313  {cpr 4608

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-v 3466  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607  df-pr 4609

This theorem is referenced by:  preqsnd  4840  0nelop  5476  fr2nr  5636  mreincl  17616  subrngin  20526  subrgin  20561  lssincl  20927  incld  22986  umgrnloopv  29090  upgr1elem  29096  usgrnloopvALT  29185  difelsiga  34169  inelpisys  34190  inidl  38059  coss0  38502  pmapmeet  39797  diameetN  41080  dihmeetlem2N  41323  dihmeetcN  41326  dihmeet  41367  infsubc  48994  infsubc2  48995