Theorem nelelne 2880

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 2879	. 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵) → 𝐶 ≠ 𝐴)
2	1	expcom 450	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977   ≠ wne 2780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-ne 2782

This theorem is referenced by:  difsn  4269  prter2  33184  frgrncvvdeqlemA  41476  frgrncvvdeqlemC  41478