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 ) ) )