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 2956  ∅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 2733

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902  df-un 3904  df-nul 4280  df-sn 4585  df-pr 4587

This theorem is used by:  preqsnd  4819  0nelop  5468  fr2nr  5628  mreincl  17769  subrngin  20813  subrgin  20848  lssincl  21240  incld  23361  umgrnloopv  29684  upgr1elem  29690  usgrnloopvALT  29782  inlidl  33971  inelpisys  34787  inidl  38964  coss0  39501  pmapmeet  40830  diameetN  42113  dihmeetlem2N  42356  dihmeetcN  42359  dihmeet  42400  infsubc  50167  infsubc2  50168