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| Description: Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.) | 
| Ref | Expression | 
|---|---|
| sotrieq2 | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sotrieq 5623 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | |
| 2 | ioran 986 | . 2 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)) | |
| 3 | 1, 2 | bitrdi 287 | 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 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 Or wor 5591 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-po 5592 df-so 5593 | 
| This theorem is referenced by: fisupg 9324 supmo 9492 infmo 9535 fiinfg 9539 lttri3 11344 xrlttri3 13185 nosupbnd1lem2 27754 noinfbnd1lem2 27769 slttrieq2 27795 weiunfrlem 36465 wessf1ornlem 45190 | 
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