Theorem ioossico

Description: An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017)

	Ref	Expression
Assertion	ioossico	⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,) 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-ioo	⊢ (,) = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ) } )
2		df-ico	⊢ [,) = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏 ) } )
3		xrltle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 → 𝐴 ≤ 𝑤 ) )
4		idd	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝑤 < 𝐵 → 𝑤 < 𝐵 ) )
5	1 2 3 4	ixxssixx	⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,) 𝐵 )