This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )