Theorem n0ii 4289

Description: If a class has elements, then it is not empty. Inference associated with n0i 4286. (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 4286	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅)
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐵 = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-dif 3902  df-nul 4280

This theorem is used by:  iin0  5327  snsn0non  6484  tfrlem16  8382  hon0  32274  dmadjrnb  32387  bnj98  35376  noinfepfnregs  35658  prv0  36009  dvnprodlem3  46776