Theorem neneor 3041

Description: If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)

Assertion

Ref	Expression
neneor	⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶))

Proof of Theorem neneor

Step	Hyp	Ref	Expression
1		eqtr3 2762	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
2	1	necon3ai 2964	. 2 ⊢ (𝐴 ≠ 𝐵 → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
3		neorian 3036	. 2 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
4	2, 3	sylibr 234	1 ⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539   ≠ wne 2939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-cleq 2728  df-ne 2940

This theorem is referenced by:  wemapso2lem  9593  trgcopyeulem  28814