Theorem iccssico2

Description: Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015)

	Ref	Expression
Assertion	iccssico2	⊢ ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ico	⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } )
2	1	elmpocl1	⊢ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) → 𝐴 ∈ ℝ* )
3	2	adantr	⊢ ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐴 ∈ ℝ* )
4	1	elmpocl2	⊢ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) → 𝐵 ∈ ℝ* )
5	4	adantr	⊢ ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐵 ∈ ℝ* )
6	1	elixx3g	⊢ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) ) )
7	6	simprbi	⊢ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) → ( 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵 ) )
8	7	simpld	⊢ ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) → 𝐴 ≤ 𝐶 )
9	8	adantr	⊢ ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐴 ≤ 𝐶 )
10	1	elixx3g	⊢ ( 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵 ) ) )
11	10	simprbi	⊢ ( 𝐷 ∈ ( 𝐴 [,) 𝐵 ) → ( 𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵 ) )
12	11	simprd	⊢ ( 𝐷 ∈ ( 𝐴 [,) 𝐵 ) → 𝐷 < 𝐵 )
13	12	adantl	⊢ ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ) → 𝐷 < 𝐵 )
14		iccssico	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,) 𝐵 ) )
15	3 5 9 13 14	syl22anc	⊢ ( ( 𝐶 ∈ ( 𝐴 [,) 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,) 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,) 𝐵 ) )