Metamath Proof Explorer
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 |
|
Hypotheses |
issod.1 |
⊢ ( 𝜑 → 𝑅 Po 𝐴 ) |
|
|
issod.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
|
Assertion |
issod |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
issod.1 |
⊢ ( 𝜑 → 𝑅 Po 𝐴 ) |
2 |
|
issod.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
3 |
2
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
4 |
|
df-so |
⊢ ( 𝑅 Or 𝐴 ↔ ( 𝑅 Po 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ) |
5 |
1 3 4
|
sylanbrc |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |