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Metamath Proof Explorer

Theorem metpsmet

Description: A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Assertion metpsmet ⊢ D ∈ Met ⁡ X → D ∈ PsMet ⁡ X

Proof

Step Hyp Ref Expression
1 metxmet ⊢ D ∈ Met ⁡ X → D ∈ ∞Met ⁡ X
2 xmetpsmet ⊢ D ∈ ∞Met ⁡ X → D ∈ PsMet ⁡ X
3 1 2 syl ⊢ D ∈ Met ⁡ X → D ∈ PsMet ⁡ X