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Mirrors > Home > MPE Home > Th. List > poltletr | Structured version Visualization version GIF version |
Description: Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
poltletr | ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poleloe 5984 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶))) | |
2 | 1 | 3ad2ant3 1127 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶))) |
3 | 2 | adantl 482 | . . 3 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶))) |
4 | 3 | anbi2d 628 | . 2 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵(𝑅 ∪ I )𝐶) ↔ (𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶)))) |
5 | potr 5479 | . . . . 5 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) | |
6 | 5 | com12 32 | . . . 4 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴𝑅𝐶)) |
7 | breq2 5061 | . . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)) | |
8 | 7 | biimpac 479 | . . . . 5 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵 = 𝐶) → 𝐴𝑅𝐶) |
9 | 8 | a1d 25 | . . . 4 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵 = 𝐶) → ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴𝑅𝐶)) |
10 | 6, 9 | jaodan 951 | . . 3 ⊢ ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶)) → ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴𝑅𝐶)) |
11 | 10 | com12 32 | . 2 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶)) → 𝐴𝑅𝐶)) |
12 | 4, 11 | sylbid 241 | 1 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 class class class wbr 5057 I cid 5452 Po wpo 5465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-po 5467 df-xp 5554 df-rel 5555 |
This theorem is referenced by: soltmin 5989 |
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