Theorem measge0

Description: A measure is nonnegative. (Contributed by Thierry Arnoux, 9-Mar-2018)

	Ref	Expression
Assertion	measge0	|- ( ( M e. ( measures ` S ) /\ A e. S ) -> 0 <_ ( M ` A ) )

Proof

Step	Hyp	Ref	Expression
1		measvxrge0	|- ( ( M e. ( measures ` S ) /\ A e. S ) -> ( M ` A ) e. ( 0 [,] +oo ) )
2		elxrge0	|- ( ( M ` A ) e. ( 0 [,] +oo ) <-> ( ( M ` A ) e. RR* /\ 0 <_ ( M ` A ) ) )
3	1 2	sylib	|- ( ( M e. ( measures ` S ) /\ A e. S ) -> ( ( M ` A ) e. RR* /\ 0 <_ ( M ` A ) ) )
4	3	simprd	|- ( ( M e. ( measures ` S ) /\ A e. S ) -> 0 <_ ( M ` A ) )