Theorem nel02 4284

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 4283	. 2 ⊢ ¬ 𝐵 ∈ ∅
2		eleq2 2849	. 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 2144  ∅c0 4278

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-dif 4047  df-nul 4279

This theorem is used by:  n0i  4285  iresn0n0  6045  0mpo0  7503  chnccat  18801  dfprlng2  29426  nbgr0vtx  29937  disjxun0  33211  noinfepregs  35850  j0ltaleph  35940  disjlem14  39909  oe0rif  44363  clnbgr0vtx  49010  iineq0  50006  nelsubclem  50251