Theorem prnzg 4730

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 4712	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2	1	ne0d 4293	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2925  ∅c0 4284  {cpr 4579

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 3438  df-dif 3906  df-un 3908  df-nul 4285  df-sn 4578  df-pr 4580

This theorem is referenced by:  preqsnd  4810  0nelop  5439  fr2nr  5596  mreincl  17501  subrngin  20446  subrgin  20481  lssincl  20868  incld  22928  umgrnloopv  29051  upgr1elem  29057  usgrnloopvALT  29146  difelsiga  34106  inelpisys  34127  inidl  38020  coss0  38466  pmapmeet  39762  diameetN  41045  dihmeetlem2N  41288  dihmeetcN  41291  dihmeet  41332  infsubc  49055  infsubc2  49056