Theorem nel02 4298

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 4296	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2901	. 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 329	1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2114  ∅c0 4291

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-dif 3939  df-nul 4292

This theorem is referenced by:  iresn0n0  5923  0mpo0  7237  disjxun0  30324