Theorem icombl1

Description: A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014)

	Ref	Expression
Assertion	icombl1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 [,) +∞ ) ∈ dom vol )

Proof

Step	Hyp	Ref	Expression
1		rexr	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
2		pnfxr	⊢ +∞ ∈ ℝ*
3	2	a1i	⊢ ( 𝐴 ∈ ℝ → +∞ ∈ ℝ* )
4		ltpnf	⊢ ( 𝐴 ∈ ℝ → 𝐴 < +∞ )
5		snunioo	⊢ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 < +∞ ) → ( { 𝐴 } ∪ ( 𝐴 (,) +∞ ) ) = ( 𝐴 [,) +∞ ) )
6	1 3 4 5	syl3anc	⊢ ( 𝐴 ∈ ℝ → ( { 𝐴 } ∪ ( 𝐴 (,) +∞ ) ) = ( 𝐴 [,) +∞ ) )
7		snssi	⊢ ( 𝐴 ∈ ℝ → { 𝐴 } ⊆ ℝ )
8		ovolsn	⊢ ( 𝐴 ∈ ℝ → ( vol* ‘ { 𝐴 } ) = 0 )
9		nulmbl	⊢ ( ( { 𝐴 } ⊆ ℝ ∧ ( vol* ‘ { 𝐴 } ) = 0 ) → { 𝐴 } ∈ dom vol )
10	7 8 9	syl2anc	⊢ ( 𝐴 ∈ ℝ → { 𝐴 } ∈ dom vol )
11		ioombl1	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 (,) +∞ ) ∈ dom vol )
12	1 11	syl	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 (,) +∞ ) ∈ dom vol )
13		unmbl	⊢ ( ( { 𝐴 } ∈ dom vol ∧ ( 𝐴 (,) +∞ ) ∈ dom vol ) → ( { 𝐴 } ∪ ( 𝐴 (,) +∞ ) ) ∈ dom vol )
14	10 12 13	syl2anc	⊢ ( 𝐴 ∈ ℝ → ( { 𝐴 } ∪ ( 𝐴 (,) +∞ ) ) ∈ dom vol )
15	6 14	eqeltrrd	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 [,) +∞ ) ∈ dom vol )