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Metamath Proof Explorer

Theorem unidmvol

Description: The union of the Lebesgue measurable sets is RR . (Contributed by Thierry Arnoux, 30-Jan-2017)

		Ref	Expression
	Assertion	unidmvol	$⊢ ⋃ dom vol = ℝ$

Step	Hyp	Ref	Expression
1		unissb	$⊢ ⋃ dom vol \subseteq ℝ \leftrightarrow \forall x \in dom vol x \subseteq ℝ$
2		mblss	$⊢ x \in dom vol \to x \subseteq ℝ$
3	1 2	mprgbir	$⊢ ⋃ dom vol \subseteq ℝ$
4		rembl	$⊢ ℝ \in dom vol$
5		unissel	$⊢ (⋃ dom vol \subseteq ℝ \land ℝ \in dom vol) \to ⋃ dom vol = ℝ$
6	3 4 5	mp2an	$⊢ ⋃ dom vol = ℝ$