Theorem nelelne 3032

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 3031	. 2 ⊢ ((𝐶 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐵) → 𝐶 ≠ 𝐴)
2	1	expcom 413	1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ≠ wne 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2728  df-clel 2810  df-ne 2934

This theorem is referenced by:  difsn  4779  feldmfvelcdm  7081  resf1extb  7935  elneq  9617  frgrncvvdeqlem7  30291  frgrncvvdeqlem9  30293  prter2  38904