Theorem iooss2

Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 3-Nov-2013)

	Ref	Expression
Assertion	iooss2	⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶 ) → ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 (,) 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
2		xrltletr	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝑤 < 𝐶 ) )
3	1 1 2	ixxss2	⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶 ) → ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 (,) 𝐶 ) )