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Metamath Proof Explorer

Theorem dmvolss

Description: Lebesgue measurable sets are subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021)

		Ref	Expression
	Assertion	dmvolss	$⊢ dom vol \subseteq 𝒫 ℝ$

Step	Hyp	Ref	Expression
1		elex	$⊢ x \in dom vol \to x \in V$
2		mblss	$⊢ x \in dom vol \to x \subseteq ℝ$
3	1 2	elpwd	$⊢ x \in dom vol \to x \in 𝒫 ℝ$
4	3	rgen	$⊢ \forall x \in dom vol x \in 𝒫 ℝ$
5		dfss3	$⊢ dom vol \subseteq 𝒫 ℝ \leftrightarrow \forall x \in dom vol x \in 𝒫 ℝ$
6	4 5	mpbir	$⊢ dom vol \subseteq 𝒫 ℝ$