Theorem n0ii 4292

Description: If a class has elements, then it is not empty. Inference associated with n0i 4289. (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 4289	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐵 = ∅

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  ∅c0 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-dif 3901  df-nul 4283

This theorem is referenced by:  iin0  5304  snsn0non  6440  tfrlem16  8321  hon0  31794  dmadjrnb  31907  bnj98  34951  prv0  35546  dvnprodlem3  46108