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Metamath Proof Explorer

Theorem metgt0

Description: The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of Gleason p. 223 and its converse. (Contributed by NM, 27-Aug-2006)

Ref Expression
Assertion metgt0 ⊢ D ∈ Met ⁡ X ∧ A ∈ X ∧ B ∈ X → A ≠ B ↔ 0 < A D B

Proof

Step Hyp Ref Expression
1 metxmet ⊢ D ∈ Met ⁡ X → D ∈ ∞Met ⁡ X
2 xmetgt0 ⊢ D ∈ ∞Met ⁡ X ∧ A ∈ X ∧ B ∈ X → A ≠ B ↔ 0 < A D B
3 1 2 syl3an1 ⊢ D ∈ Met ⁡ X ∧ A ∈ X ∧ B ∈ X → A ≠ B ↔ 0 < A D B