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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 5754 | . . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
3 | 2 | simprd 495 | . . 3 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ 𝑆) |
4 | soi.1 | . . . . . . 7 ⊢ 𝑅 Or 𝑆 | |
5 | sotric 5626 | . . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶))) | |
6 | 4, 5 | mpan 690 | . . . . . 6 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶))) |
7 | 6 | con2bid 354 | . . . . 5 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵)) |
8 | breq2 5152 | . . . . . . 7 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐴𝑅𝐵)) | |
9 | 8 | biimprd 248 | . . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
10 | 4, 1 | sotri 6150 | . . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
11 | 10 | expcom 413 | . . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
12 | 9, 11 | jaoi 857 | . . . . 5 ⊢ ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)) |
13 | 7, 12 | biimtrrdi 254 | . . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))) |
14 | 13 | com3r 87 | . . 3 ⊢ (𝐴𝑅𝐵 → ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶))) |
15 | 3, 14 | mpan2d 694 | . 2 ⊢ (𝐴𝑅𝐵 → (𝐶 ∈ 𝑆 → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶))) |
16 | 15 | 3imp21 1113 | 1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 Or wor 5596 × cxp 5687 |
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 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
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-ral 3060 df-rex 3069 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-opab 5211 df-po 5597 df-so 5598 df-xp 5695 |
This theorem is referenced by: archnq 11018 |
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