Theorem volge0

Description: The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	volge0	$$⊢A\in \mathrm{dom}vol\to 0\le vol\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		mblss	$$⊢A\in \mathrm{dom}vol\to A\subseteq ℝ$$
2		ovolge0	$$⊢A\subseteq ℝ\to 0\le {vol}^{*}\left(A\right)$$
3	1 2	syl	$$⊢A\in \mathrm{dom}vol\to 0\le {vol}^{*}\left(A\right)$$
4		mblvol	$$⊢A\in \mathrm{dom}vol\to vol\left(A\right)={vol}^{*}\left(A\right)$$
5	3 4	breqtrrd	$$⊢A\in \mathrm{dom}vol\to 0\le vol\left(A\right)$$