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Mirrors > Home > MPE Home > Th. List > sotri3 | Structured version Visualization version GIF version |
Description: A transitivity relation. (Read 𝐴 < 𝐵 and 𝐵 ≤ 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
soi.1 | ⊢ 𝑅 Or 𝑆 |
soi.2 | ⊢ 𝑅 ⊆ (𝑆 × 𝑆) |
Ref | Expression |
---|---|
sotri3 | ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | soi.2 | . . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
2 | 1 | brel 5369 | . . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
3 | 2 | simprd 490 | . . 3 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ 𝑆) |
4 | soi.1 | . . . . . . 7 ⊢ 𝑅 Or 𝑆 | |
5 | sotric 5257 | . . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶))) | |
6 | 4, 5 | mpan 682 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶))) |
7 | 6 | con2bid 346 | . . . . 5 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵)) |
8 | breq2 4845 | . . . . . . 7 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐴𝑅𝐵)) | |
9 | 8 | biimprd 240 | . . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
10 | 4, 1 | sotri 5739 | . . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
11 | 10 | expcom 403 | . . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
12 | 9, 11 | jaoi 884 | . . . . 5 ⊢ ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
13 | 7, 12 | syl6bir 246 | . . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))) |
14 | 13 | com3r 87 | . . 3 ⊢ (𝐴𝑅𝐵 → ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶))) |
15 | 3, 14 | mpan2d 686 | . 2 ⊢ (𝐴𝑅𝐵 → (𝐶 ∈ 𝑆 → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶))) |
16 | 15 | 3imp21 1142 | 1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 ∨ wo 874 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ⊆ wss 3767 class class class wbr 4841 Or wor 5230 × cxp 5308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-po 5231 df-so 5232 df-xp 5316 |
This theorem is referenced by: archnq 10088 |
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