Theorem nel02 4285

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 4284	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2850	. 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅))
3	1, 2	mtbiri 330	1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = 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 2733

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 2740  df-cleq 2753  df-clel 2836  df-dif 3902  df-nul 4280

This theorem is used by:  n0i  4286  iresn0n0  6046  0mpo0  7503  chnccat  18800  dfprlng2  29425  nbgr0vtx  29936  disjxun0  33168  noinfepregs  35801  disjlem14  39833  oe0rif  44286  clnbgr0vtx  48933  iineq0  49929  nelsubclem  50174