MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prnzg Structured version   Visualization version   GIF version

Theorem prnzg 4342
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 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Proof of Theorem prnzg
StepHypRef Expression
1 prid1g 4327 . 2 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 ne0i 3954 . 2 (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅)
31, 2syl 17 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wne 2823  c0 3948  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  0nelop  4989  fr2nr  5121  mreincl  16306  subrgin  18851  lssincl  19013  incld  20895  umgrnloopv  26046  upgr1elem  26052  usgrnloopvALT  26138  difelsiga  30324  inelpisys  30345  inidl  33959  coss0  34369  pmapmeet  35377  diameetN  36662  dihmeetlem2N  36905  dihmeetcN  36908  dihmeet  36949
  Copyright terms: Public domain W3C validator