Theorem neneor 3088

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 2820	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
2	1	necon3ai 3011	. 2 ⊢ (𝐴 ≠ 𝐵 → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
3		neorian 3081	. 2 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
4	2, 3	sylibr 235	1 ⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 842   = wceq 1525   ≠ wne 2986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-9 2093  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-cleq 2790  df-ne 2987

This theorem is referenced by:  wemapso2lem  8869  trgcopyeulem  26277