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Mirrors > Home > MPE Home > Th. List > sotri2 | 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 |
---|---|
sotri2 | ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | soi.2 | . . . . . 6 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
2 | 1 | brel 5090 | . . . . 5 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) |
3 | 2 | simpld 474 | . . . 4 ⊢ (𝐵𝑅𝐶 → 𝐵 ∈ 𝑆) |
4 | soi.1 | . . . . . . . 8 ⊢ 𝑅 Or 𝑆 | |
5 | sotric 4985 | . . . . . . . 8 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵))) | |
6 | 4, 5 | mpan 702 | . . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵))) |
7 | 6 | con2bid 343 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴)) |
8 | breq1 4586 | . . . . . . . 8 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 ↔ 𝐴𝑅𝐶)) | |
9 | 8 | biimpd 218 | . . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)) |
10 | 4, 1 | sotri 5442 | . . . . . . . 8 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
11 | 10 | ex 449 | . . . . . . 7 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)) |
12 | 9, 11 | jaoi 393 | . . . . . 6 ⊢ ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)) |
13 | 7, 12 | syl6bir 243 | . . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))) |
14 | 13 | com3r 85 | . . . 4 ⊢ (𝐵𝑅𝐶 → ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶))) |
15 | 3, 14 | mpand 707 | . . 3 ⊢ (𝐵𝑅𝐶 → (𝐴 ∈ 𝑆 → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶))) |
16 | 15 | com3l 87 | . 2 ⊢ (𝐴 ∈ 𝑆 → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))) |
17 | 16 | 3imp 1249 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 Or wor 4958 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-po 4959 df-so 4960 df-xp 5044 |
This theorem is referenced by: supsrlem 9811 |
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