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Theorem n0ii 4152
 Description: If a class has elements, then it is not empty. Inference associated with n0i 4149. (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 4149 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
31, 2ax-mp 5 1 ¬ 𝐵 = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1658   ∈ wcel 2166  ∅c0 4144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-nul 4145 This theorem is referenced by:  iin0  5061  snsn0non  6081  tfrlem16  7755  pwcdadom  9353  nnunb  11614  hon0  29207  dmadjrnb  29320  bnj98  31483  dvnprodlem3  40958
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