Theorem icossico

Description: Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017)

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

Proof

Step	Hyp	Ref	Expression
1		df-ico	⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } )
2		xrletr	⊢ ( ( 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 ) )