Theorem nel02 4345

Description: The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)

Assertion

Ref	Expression
nel02	⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)

Proof of Theorem nel02

Step	Hyp	Ref	Expression
1		noel 4343	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2818	. 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 326	1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ∈ wcel 2100  ∅c0 4335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-dif 3962  df-nul 4336

This theorem is referenced by:  iresn0n0  6068  0mpo0  7514  nbgr0vtx  29314  disjxun0  32520  disjlem14  38534  oe0rif  43013  clnbgr0vtx  47467