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

Theorem n0ii 4275
 Description: If a class has elements, then it is not empty. Inference associated with n0i 4272. (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 4272 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
31, 2ax-mp 5 1 ¬ 𝐵 = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2115  ∅c0 4266 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-dif 3913  df-nul 4267 This theorem is referenced by:  iin0  5234  snsn0non  6282  tfrlem16  8004  hon0  29555  dmadjrnb  29668  bnj98  32147  prv0  32685  bj-isrvec  34593  dvnprodlem3  42409
 Copyright terms: Public domain W3C validator