Theorem nel02 4288

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 4286	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2853	. 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 1569   ∈ wcel 2146  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-dif 4050  df-nul 4282

This theorem is used by:  n0i  4289  iresn0n0  6052  0mpo0  7513  chnccat  18815  dfprlng2  29440  nbgr0vtx  29951  disjxun0  33225  noinfepregs  35875  j0ltaleph  35965  jltaleph  35983  disjlem14  39976  oe0rif  44430  clnbgr0vtx  49076  iineq0  50073  nelsubclem  50323