Theorem iooss1

Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 20-Feb-2015)

	Ref	Expression
Assertion	iooss1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 (,) 𝐶 ) ⊆ ( 𝐴 (,) 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
2		xrlelttr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝑤 ) → 𝐴 < 𝑤 ) )
3	1 1 2	ixxss1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 (,) 𝐶 ) ⊆ ( 𝐴 (,) 𝐶 ) )