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	$$⊢\phi \to X\in \mathrm{Fin}$$
	voncl.s	$$⊢S=\mathrm{dom}\mathrm{voln}\left(X\right)$$
	voncl.a	$$⊢\phi \to A\in S$$
Assertion	voncl	$$⊢\phi \to \mathrm{voln}\left(X\right)\left(A\right)\in \left[0,\mathrm{+\infty }\right]$$

Proof

Step	Hyp	Ref	Expression
1		voncl.x	$$⊢\phi \to X\in \mathrm{Fin}$$
2		voncl.s	$$⊢S=\mathrm{dom}\mathrm{voln}\left(X\right)$$
3		voncl.a	$$⊢\phi \to A\in S$$
4	1	vonmea	$$⊢\phi \to \mathrm{voln}\left(X\right)\in \mathrm{Meas}$$
5	4 2 3	meacl	$$⊢\phi \to \mathrm{voln}\left(X\right)\left(A\right)\in \left[0,\mathrm{+\infty }\right]$$