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Mirrors > Home > MPE Home > Th. List > sotrieq | Structured version Visualization version GIF version |
Description: Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
sotrieq | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sonr 5498 | . . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | |
2 | 1 | adantrr 715 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵) |
3 | pm1.2 900 | . . . . . 6 ⊢ ((𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) → 𝐵𝑅𝐵) | |
4 | 2, 3 | nsyl 142 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵)) |
5 | breq2 5072 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶)) | |
6 | breq1 5071 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵𝑅𝐵 ↔ 𝐶𝑅𝐵)) | |
7 | 5, 6 | orbi12d 915 | . . . . . 6 ⊢ (𝐵 = 𝐶 → ((𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
8 | 7 | notbid 320 | . . . . 5 ⊢ (𝐵 = 𝐶 → (¬ (𝐵𝑅𝐵 ∨ 𝐵𝑅𝐵) ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
9 | 4, 8 | syl5ibcom 247 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 → ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
10 | 9 | con2d 136 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) → ¬ 𝐵 = 𝐶)) |
11 | solin 5500 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) | |
12 | 3orass 1086 | . . . . . 6 ⊢ ((𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) | |
13 | 11, 12 | sylib 220 | . . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) |
14 | or12 917 | . . . . 5 ⊢ ((𝐵𝑅𝐶 ∨ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) ↔ (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | |
15 | 13, 14 | sylib 220 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ∨ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
16 | 15 | ord 860 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (¬ 𝐵 = 𝐶 → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) |
17 | 10, 16 | impbid 214 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ ¬ 𝐵 = 𝐶)) |
18 | 17 | con2bid 357 | 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 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 Or wor 5475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-po 5476 df-so 5477 |
This theorem is referenced by: sotrieq2 5505 sossfld 6045 soisores 7082 soisoi 7083 weniso 7109 wemapsolem 9016 distrlem4pr 10450 addcanpr 10470 sqgt0sr 10530 lttri2 10725 xrlttri2 12538 xrltne 12559 sotrine 33005 soseq 33098 |
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