Theorem omexrcl

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

	Ref	Expression
Hypotheses	omexrcl.o	⊢ ( 𝜑 → 𝑂 ∈ OutMeas )
	omexrcl.x	⊢ 𝑋 = ∪ dom 𝑂
	omexrcl.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝑋 )
Assertion	omexrcl	⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		omexrcl.o	⊢ ( 𝜑 → 𝑂 ∈ OutMeas )
2		omexrcl.x	⊢ 𝑋 = ∪ dom 𝑂
3		omexrcl.a	⊢ ( 𝜑 → 𝐴 ⊆ 𝑋 )
4		iccssxr	⊢ ( 0 [,] +∞ ) ⊆ ℝ*
5	1 2 3	omecl	⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
6	4 5	sseldi	⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) ∈ ℝ* )