Theorem prnzg 4738

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 4720	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2	1	ne0d 4301	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2925  ∅c0 4292  {cpr 4587

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3446  df-dif 3914  df-un 3916  df-nul 4293  df-sn 4586  df-pr 4588

This theorem is referenced by:  preqsnd  4819  0nelop  5451  fr2nr  5608  mreincl  17536  subrngin  20481  subrgin  20516  lssincl  20903  incld  22963  umgrnloopv  29086  upgr1elem  29092  usgrnloopvALT  29181  difelsiga  34116  inelpisys  34137  inidl  38017  coss0  38463  pmapmeet  39760  diameetN  41043  dihmeetlem2N  41286  dihmeetcN  41289  dihmeet  41330  infsubc  49042  infsubc2  49043