Theorem ioossioc

Description: An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	ioossioc	⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 (,] 𝐵 )

Proof

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