Theorem omexrcl

Description: The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	omexrcl.o	|- ( ph -> O e. OutMeas )
	omexrcl.x	|- X = U. dom O
	omexrcl.a	|- ( ph -> A C_ X )
Assertion	omexrcl	|- ( ph -> ( O ` A ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		omexrcl.o	|- ( ph -> O e. OutMeas )
2		omexrcl.x	|- X = U. dom O
3		omexrcl.a	|- ( ph -> A C_ X )
4		iccssxr	|- ( 0 [,] +oo ) C_ RR*
5	1 2 3	omecl	|- ( ph -> ( O ` A ) e. ( 0 [,] +oo ) )
6	4 5	sseldi	|- ( ph -> ( O ` A ) e. RR* )