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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 2643 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | |
2 | 1 | necon3ai 2819 | . 2 ⊢ (𝐴 ≠ 𝐵 → ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) |
3 | neorian 2888 | . 2 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) | |
4 | 2, 3 | sylibr 224 | 1 ⊢ (𝐴 ≠ 𝐵 → (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ≠ wne 2794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-cleq 2615 df-ne 2795 |
This theorem is referenced by: trgcopyeulem 25697 |
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