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

Proof

Step	Hyp	Ref	Expression
1		unissb	⊢ ( ∪ dom vol ⊆ ℝ ↔ ∀ 𝑥 ∈ dom vol 𝑥 ⊆ ℝ )
2		mblss	⊢ ( 𝑥 ∈ dom vol → 𝑥 ⊆ ℝ )
3	1 2	mprgbir	⊢ ∪ dom vol ⊆ ℝ
4		rembl	⊢ ℝ ∈ dom vol
5		unissel	⊢ ( ( ∪ dom vol ⊆ ℝ ∧ ℝ ∈ dom vol ) → ∪ dom vol = ℝ )
6	3 4 5	mp2an	⊢ ∪ dom vol = ℝ