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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 485 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) |
3 | issoi.2 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
4 | 3 | adantl 485 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
5 | 2, 4 | ispod 5446 | . . 3 ⊢ (⊤ → 𝑅 Po 𝐴) |
6 | issoi.3 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) | |
7 | 6 | adantl 485 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
8 | 5, 7 | issod 5470 | . 2 ⊢ (⊤ → 𝑅 Or 𝐴) |
9 | 8 | mptru 1545 | 1 ⊢ 𝑅 Or 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ w3o 1083 ∧ w3a 1084 ⊤wtru 1539 ∈ wcel 2111 class class class wbr 5030 Or wor 5437 |
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 1911 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 df-tru 1541 df-ral 3111 df-po 5438 df-so 5439 |
This theorem is referenced by: isso2i 5472 ltsopr 10443 sltsolem1 33293 |
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