Theorem volge0

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

	Ref	Expression
Assertion	volge0	|- ( A e. dom vol -> 0 <_ ( vol ` A ) )

Proof

Step	Hyp	Ref	Expression
1		mblss	|- ( A e. dom vol -> A C_ RR )
2		ovolge0	|- ( A C_ RR -> 0 <_ ( vol* ` A ) )
3	1 2	syl	|- ( A e. dom vol -> 0 <_ ( vol* ` A ) )
4		mblvol	|- ( A e. dom vol -> ( vol ` A ) = ( vol* ` A ) )
5	3 4	breqtrrd	|- ( A e. dom vol -> 0 <_ ( vol ` A ) )