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Mirrors > Home > MPE Home > Th. List > issoi | Structured version Visualization version GIF version |
Description: An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
issoi.1 | ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥) |
issoi.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
issoi.3 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
issoi | ⊢ 𝑅 Or 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issoi.1 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) |
3 | issoi.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
4 | 3 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
5 | 2, 4 | ispod 5554 | . . 3 ⊢ (⊤ → 𝑅 Po 𝐴) |
6 | issoi.3 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) | |
7 | 6 | adantl 482 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
8 | 5, 7 | issod 5578 | . 2 ⊢ (⊤ → 𝑅 Or 𝐴) |
9 | 8 | mptru 1548 | 1 ⊢ 𝑅 Or 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ w3o 1086 ∧ w3a 1087 ⊤wtru 1542 ∈ wcel 2106 class class class wbr 5105 Or wor 5544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ral 3065 df-po 5545 df-so 5546 |
This theorem is referenced by: isso2i 5580 ltsopr 10968 sltsolem1 27023 |
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