Theorem icossicc

Description: A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016)

	Ref	Expression
Assertion	icossicc	⊢ ( 𝐴 [,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-ico	⊢ [,) = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏 ) } )
2		df-icc	⊢ [,] = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 ≤ 𝑥 ∧ 𝑥 ≤ 𝑏 ) } )
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 )