Theorem voncl

Description: The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	voncl.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
	voncl.s	⊢ 𝑆 = dom ( voln ‘ 𝑋 )
	voncl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
Assertion	voncl	⊢ ( 𝜑 → ( ( voln ‘ 𝑋 ) ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		voncl.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
2		voncl.s	⊢ 𝑆 = dom ( voln ‘ 𝑋 )
3		voncl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
4	1	vonmea	⊢ ( 𝜑 → ( voln ‘ 𝑋 ) ∈ Meas )
5	4 2 3	meacl	⊢ ( 𝜑 → ( ( voln ‘ 𝑋 ) ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )