Theorem nelelne 3059

Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)

Assertion

Ref	Expression
nelelne	⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))

Proof of Theorem nelelne

Step	Hyp	Ref	Expression
1		nelne2 3056	. 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵) → 𝐶 ≠ 𝐴)
2	1	expcom 418	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2143   ≠ wne 2958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-clel 2838  df-ne 2959

This theorem is referenced by:  difsn  4766  feldmfvelcdm  7081  resf1extb  7927  elneq  9559  frgrncvvdeqlem7  30656  frgrncvvdeqlem9  30658  prter2  39655