Theorem elcarsgss

Description: Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020)

	Ref	Expression
Hypotheses	carsgval.1	|- ( ph -> O e. V )
	carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
	difelcarsg.1	|- ( ph -> A e. ( toCaraSiga ` M ) )
Assertion	elcarsgss	|- ( ph -> A C_ O )

Proof

Step	Hyp	Ref	Expression
1		carsgval.1	|- ( ph -> O e. V )
2		carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
3		difelcarsg.1	|- ( ph -> A e. ( toCaraSiga ` M ) )
4	1 2	carsgcl	|- ( ph -> ( toCaraSiga ` M ) C_ ~P O )
5	4 3	sseldd	|- ( ph -> A e. ~P O )
6	5	elpwid	|- ( ph -> A C_ O )