Theorem ioossioo

Description: Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017)

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

Proof

Step	Hyp	Ref	Expression
1		df-ioo	⊢ (,) = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ) } )
2		xrlelttr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑤 ) → 𝐴 < 𝑤 ) )
3		xrltletr	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝑤 < 𝐷 ∧ 𝐷 ≤ 𝐵 ) → 𝑤 < 𝐵 ) )
4	1 1 2 3	ixxss12	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵 ) ) → ( 𝐶 (,) 𝐷 ) ⊆ ( 𝐴 (,) 𝐵 ) )