Theorem nel02 4292

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 4291	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2852	. 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 329	1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1561   ∈ wcel 2143  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-dif 3908  df-nul 4287

This theorem is referenced by:  n0i  4293  iresn0n0  6044  0mpo0  7480  chnccat  18659  nbgr0vtx  29557  disjxun0  32775  noinfepregs  35430  disjlem14  39401  oe0rif  43863  clnbgr0vtx  48459  iineq0  49442  nelsubclem  49689