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

Metamath Proof Explorer

Theorem volioo

Description: The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017)

		Ref	Expression
	Assertion	volioo	$⊢ (A \in ℝ \land B \in ℝ \land A \leq B) \to vol ((A, B)) = B - A$

Step	Hyp	Ref	Expression
1		ioombl	$⊢ (A, B) \in dom vol$
2		mblvol	$⊢ (A, B) \in dom vol \to vol ((A, B)) = {vol}^{*} ((A, B))$
3	1 2	ax-mp	$⊢ vol ((A, B)) = {vol}^{*} ((A, B))$
4		ovolioo	$⊢ (A \in ℝ \land B \in ℝ \land A \leq B) \to {vol}^{*} ((A, B)) = B - A$
5	3 4	eqtrid	$⊢ (A \in ℝ \land B \in ℝ \land A \leq B) \to vol ((A, B)) = B - A$