Theorem n0ii 4305

Description: If a class has elements, then it is not empty. Inference associated with n0i 4302. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
n0ii.1	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
n0ii	⊢ ¬ 𝐵 = ∅

Proof of Theorem n0ii

Step	Hyp	Ref	Expression
1		n0ii.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		n0i 4302	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐵 = ∅

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-dif 3942  df-nul 4295

This theorem is referenced by:  iin0  5264  snsn0non  6312  tfrlem16  8032  hon0  29573  dmadjrnb  29686  bnj98  32143  prv0  32681  bj-isrvec  34579  dvnprodlem3  42239