This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem volge0

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

		Ref	Expression
	Assertion	volge0	$⊢ A \in dom vol \to 0 \leq vol (A)$

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