Theorem volsn

Description: A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	volsn	|- ( A e. RR -> ( vol ` { A } ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		snmbl	|- ( A e. RR -> { A } e. dom vol )
2		mblvol	|- ( { A } e. dom vol -> ( vol ` { A } ) = ( vol* ` { A } ) )
3	1 2	syl	|- ( A e. RR -> ( vol ` { A } ) = ( vol* ` { A } ) )
4		ovolsn	|- ( A e. RR -> ( vol* ` { A } ) = 0 )
5	3 4	eqtrd	|- ( A e. RR -> ( vol ` { A } ) = 0 )