Theorem meaxrcl

Description: The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	meaxrcl.1	⊢ ( 𝜑 → 𝑀 ∈ Meas )
	meaxrcl.2	⊢ 𝑆 = dom 𝑀
	meaxrcl.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
Assertion	meaxrcl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		meaxrcl.1	⊢ ( 𝜑 → 𝑀 ∈ Meas )
2		meaxrcl.2	⊢ 𝑆 = dom 𝑀
3		meaxrcl.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
4		iccssxr	⊢ ( 0 [,] +∞ ) ⊆ ℝ*
5	1 2 3	meacl	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
6	4 5	sseldi	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* )