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Mirrors > Home > MPE Home > Th. List > sotrine | Structured version Visualization version GIF version |
Description: Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
sotrine | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ≠ 𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sotrieq 5627 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | |
2 | 1 | bicomd 223 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ 𝐵 = 𝐶)) |
3 | 2 | necon1abid 2977 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ≠ 𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 Or wor 5596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-po 5597 df-so 5598 |
This theorem is referenced by: nosepne 27740 nosepdm 27744 slttrine 27811 |
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