Theorem measge0

Description: A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018)

	Ref	Expression
Assertion	measge0	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → 0 ≤ ( 𝑀 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		measvxrge0	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
2		elxrge0	⊢ ( ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) )
3	1 2	sylib	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) )
4	3	simprd	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → 0 ≤ ( 𝑀 ‘ 𝐴 ) )