Theorem prnzg 4739

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 4721	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
2	1	ne0d 4288	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2955  ∅c0 4279  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 3902  df-un 3904  df-nul 4280  df-sn 4585  df-pr 4587

This theorem is used by:  preqsnd  4819  0nelop  5473  fr2nr  5632  mreincl  17686  subrngin  20726  subrgin  20761  lssincl  21152  incld  23271  umgrnloopv  29566  upgr1elem  29572  usgrnloopvALT  29664  inlidl  33852  inelpisys  34668  inidl  38783  coss0  39320  pmapmeet  40649  diameetN  41932  dihmeetlem2N  42175  dihmeetcN  42178  dihmeet  42219  infsubc  49989  infsubc2  49990