Theorem neneor 3066

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 2791	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
2	1	necon3ai 2989	. 2 ⊢ (𝐴 ≠ 𝐵 → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
3		neorian 3059	. 2 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
4	2, 3	sylibr 237	1 ⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ≠ wne 2964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-cleq 2761  df-ne 2965

This theorem is referenced by:  wemapso2lem  9513  trgcopyeulem  29072