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Mirrors > Home > MPE Home > Th. List > issod | 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 |
---|---|
issod.1 | ⊢ (𝜑 → 𝑅 Po 𝐴) |
issod.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
issod | ⊢ (𝜑 → 𝑅 Or 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issod.1 | . 2 ⊢ (𝜑 → 𝑅 Po 𝐴) | |
2 | issod.2 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) | |
3 | 2 | ralrimivva 2954 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
4 | df-so 4960 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
5 | 1, 3, 4 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝑅 Or 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∨ w3o 1030 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 Po wpo 4957 Or wor 4958 |
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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ral 2901 df-so 4960 |
This theorem is referenced by: issoi 4990 swoso 7662 wemapsolem 8338 legso 25294 socnv 30908 fin2so 32566 |
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