Theorem nel02 4332

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 4330	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2823	. 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 327	1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-dif 3951  df-nul 4323

This theorem is referenced by:  iresn0n0  6052  0mpo0  7489  disjxun0  31793  disjlem14  37657  oe0rif  42021