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 ) ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴𝐵 ↔ 0 < ( 𝐴 𝐷 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 metxmet ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
2 xmetgt0 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴𝐵 ↔ 0 < ( 𝐴 𝐷 𝐵 ) ) )
3 1 2 syl3an1 ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴𝐵 ↔ 0 < ( 𝐴 𝐷 𝐵 ) ) )