MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0ii Structured version   Visualization version   GIF version

Theorem n0ii 4337
Description: If a class has elements, then it is not empty. Inference associated with n0i 4334. (Contributed by BJ, 15-Jul-2021.)
Hypothesis
Ref Expression
n0ii.1 𝐴𝐵
Assertion
Ref Expression
n0ii ¬ 𝐵 = ∅

Proof of Theorem n0ii
StepHypRef Expression
1 n0ii.1 . 2 𝐴𝐵
2 n0i 4334 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
31, 2ax-mp 5 1 ¬ 𝐵 = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-dif 3950  df-nul 4324
This theorem is referenced by:  iin0  5362  snsn0non  6494  tfrlem16  8413  hon0  31602  dmadjrnb  31715  bnj98  34498  prv0  35040  dvnprodlem3  45336
  Copyright terms: Public domain W3C validator