Theorem n0ii 4349

Description: If a class has elements, then it is not empty. Inference associated with n0i 4346. (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 4346	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐵 = ∅

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-nul 4340

This theorem is referenced by:  iin0  5368  snsn0non  6511  tfrlem16  8432  hon0  31822  dmadjrnb  31935  bnj98  34860  prv0  35415  dvnprodlem3  45904