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Metamath Proof Explorer

Theorem sopo

Description: A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997)

Ref Expression
Assertion sopo ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )

Proof

Step Hyp Ref Expression
1 df-so ⊢ ( 𝑅 Or 𝐴 ↔ ( 𝑅 Po 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) )
2 1 simplbi ⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )