Theorem iccss2

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	iccss2	⊢ ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
2	1	elixx3g	⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
3	2	simplbi	⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) )
4	3	adantr	⊢ ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) )
5	4	simp1d	⊢ ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ* )
6	4	simp2d	⊢ ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ* )
7	2	simprbi	⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
8	7	adantr	⊢ ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
9	8	simpld	⊢ ( ( 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		xrletr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝑤 ) → 𝐴 ≤ 𝑤 ) )
15		xrletr	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝑤 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵 ) → 𝑤 ≤ 𝐵 ) )
16	1 1 14 15	ixxss12	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,] 𝐵 ) )
17	5 6 9 13 16	syl22anc	⊢ ( ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐷 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 [,] 𝐷 ) ⊆ ( 𝐴 [,] 𝐵 ) )