Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > neneor | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
neneor | ⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2843 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | |
2 | 1 | necon3ai 3041 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
3 | neorian 3111 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) | |
4 | 2, 3 | sylibr 235 | 1 ⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ≠ wne 3016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-cleq 2814 df-ne 3017 |
This theorem is referenced by: wemapso2lem 9005 trgcopyeulem 26519 |
Copyright terms: Public domain | W3C validator |