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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 6025 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶))) | |
2 | 1 | 3ad2ant3 1133 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵(𝑅 ∪ I )𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶))) |
3 | 2 | adantl 481 | . . 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 5507 | . . . . 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 5074 | . . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)) | |
8 | 7 | biimpac 478 | . . . . 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 954 | . . 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 239 | 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 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∪ cun 3881 class class class wbr 5070 I cid 5479 Po wpo 5492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-id 5480 df-po 5494 df-xp 5586 df-rel 5587 |
This theorem is referenced by: soltmin 6030 |
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