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Metamath Proof Explorer

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 ) )