File: Mathlib/Order/Interval/Set/Disjoint.lean

full_name: Set.Iic_disjoint_Ioi
code: @[simp]
theorem Iic_disjoint_Ioi (h : a ≤ b) : Disjoint (Iic a) (Ioi b) 
goal: Disjoint (Set.Iic a) (Set.Ioi b)

full_name: Set.Iio_disjoint_Ici
code: @[simp]
theorem Iio_disjoint_Ici (h : a ≤ b) : Disjoint (Iio a) (Ici b) 
goal: Disjoint (Set.Iio a) (Set.Ici b)

full_name: Set.Iic_disjoint_Ioc
code: @[simp]
theorem Iic_disjoint_Ioc (h : a ≤ b) : Disjoint (Iic a) (Ioc b c) 
goal: Disjoint (Set.Iic a) (Set.Ioc b c)

full_name: Set.Ioc_disjoint_Ioc_same
code: @[simp]
theorem Ioc_disjoint_Ioc_same : Disjoint (Ioc a b) (Ioc b c) 
goal: Disjoint (Set.Ioc a b) (Set.Ioc b c)

full_name: Set.Ico_disjoint_Ico_same
code: @[simp]
theorem Ico_disjoint_Ico_same : Disjoint (Ico a b) (Ico b c) 
goal: Disjoint (Set.Ico a b) (Set.Ico b c)

full_name: Set.Ici_disjoint_Iic
code: @[simp]
theorem Ici_disjoint_Iic : Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b 
goal: Disjoint (Set.Ici a) (Set.Iic b) ↔ ¬a ≤ b

full_name: Set.Iic_disjoint_Ici
code: @[simp]
theorem Iic_disjoint_Ici : Disjoint (Iic a) (Ici b) ↔ ¬b ≤ a 
goal: Disjoint (Set.Iic a) (Set.Ici b) ↔ ¬b ≤ a

full_name: Set.Ioc_disjoint_Ioi
code: @[simp]
theorem Ioc_disjoint_Ioi (h : b ≤ c) : Disjoint (Ioc a b) (Ioi c) 
goal: Disjoint (Set.Ioc a b) (Set.Ioi c)

full_name: Set.Ioc_disjoint_Ioi_same
code: theorem Ioc_disjoint_Ioi_same : Disjoint (Ioc a b) (Ioi b) 
goal: Disjoint (Set.Ioc a b) (Set.Ioi b)

full_name: Set.iUnion_Iic
code: @[simp]
theorem iUnion_Iic : ⋃ a : α, Iic a = univ 
goal: ⋃ a, Set.Iic a = Set.univ

full_name: Set.iUnion_Ici
code: @[simp]
theorem iUnion_Ici : ⋃ a : α, Ici a = univ 
goal: ⋃ a, Set.Ici a = Set.univ

full_name: Set.iUnion_Icc_right
code: @[simp]
theorem iUnion_Icc_right (a : α) : ⋃ b, Icc a b = Ici a 
goal: ⋃ b, Set.Icc a b = Set.Ici a

full_name: Set.iUnion_Ioc_right
code: @[simp]
theorem iUnion_Ioc_right (a : α) : ⋃ b, Ioc a b = Ioi a 
goal: ⋃ b, Set.Ioc a b = Set.Ioi a

full_name: Set.iUnion_Icc_left
code: @[simp]
theorem iUnion_Icc_left (b : α) : ⋃ a, Icc a b = Iic b 
goal: ⋃ a, Set.Icc a b = Set.Iic b

full_name: Set.iUnion_Ico_left
code: @[simp]
theorem iUnion_Ico_left (b : α) : ⋃ a, Ico a b = Iio b 
goal: ⋃ a, Set.Ico a b = Set.Iio b

full_name: Set.iUnion_Iio
code: @[simp]
theorem iUnion_Iio [NoMaxOrder α] : ⋃ a : α, Iio a = univ 
goal: ⋃ a, Set.Iio a = Set.univ

full_name: Set.iUnion_Ioi
code: @[simp]
theorem iUnion_Ioi [NoMinOrder α] : ⋃ a : α, Ioi a = univ 
goal: ⋃ a, Set.Ioi a = Set.univ

full_name: Set.iUnion_Ico_right
code: @[simp]
theorem iUnion_Ico_right [NoMaxOrder α] (a : α) : ⋃ b, Ico a b = Ici a 
goal: ⋃ b, Set.Ico a b = Set.Ici a

full_name: Set.iUnion_Ioo_right
code: @[simp]
theorem iUnion_Ioo_right [NoMaxOrder α] (a : α) : ⋃ b, Ioo a b = Ioi a 
goal: ⋃ b, Set.Ioo a b = Set.Ioi a

full_name: Set.iUnion_Ioc_left
code: @[simp]
theorem iUnion_Ioc_left [NoMinOrder α] (b : α) : ⋃ a, Ioc a b = Iic b 
goal: ⋃ a, Set.Ioc a b = Set.Iic b

full_name: Set.iUnion_Ioo_left
code: @[simp]
theorem iUnion_Ioo_left [NoMinOrder α] (b : α) : ⋃ a, Ioo a b = Iio b 
goal: ⋃ a, Set.Ioo a b = Set.Iio b

full_name: Set.Ico_disjoint_Ico
code: @[simp]
theorem Ico_disjoint_Ico : Disjoint (Ico a₁ a₂) (Ico b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ 
goal: Disjoint (Set.Ico a₁ a₂) (Set.Ico b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁

full_name: Set.Ioc_disjoint_Ioc
code: @[simp]
theorem Ioc_disjoint_Ioc : Disjoint (Ioc a₁ a₂) (Ioc b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ 
goal: Disjoint (Set.Ioc a₁ a₂) (Set.Ioc b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁

full_name: Set.Ioo_disjoint_Ioo
code: @[simp]
theorem Ioo_disjoint_Ioo [DenselyOrdered α] :
    Disjoint (Set.Ioo a₁ a₂) (Set.Ioo b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ 
goal: Disjoint (Set.Ioo a₁ a₂) (Set.Ioo b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁

full_name: Set.eq_of_Ico_disjoint
code: /-- If two half-open intervals are disjoint and the endpoint of one lies in the other,
  then it must be equal to the endpoint of the other. -/
theorem eq_of_Ico_disjoint {x₁ x₂ y₁ y₂ : α} (h : Disjoint (Ico x₁ x₂) (Ico y₁ y₂)) (hx : x₁ < x₂)
    (h2 : x₂ ∈ Ico y₁ y₂) : y₁ = x₂ 
goal: y₁ = x₂

full_name: Set.iUnion_Ico_eq_Iio_self_iff
code: @[simp]
theorem iUnion_Ico_eq_Iio_self_iff {f : ι → α} {a : α} :
    ⋃ i, Ico (f i) a = Iio a ↔ ∀ x < a, ∃ i, f i ≤ x 
goal: ⋃ i, Set.Ico (f i) a = Set.Iio a ↔ ∀ x < a, ∃ i, f i ≤ x

full_name: Set.iUnion_Ioc_eq_Ioi_self_iff
code: @[simp]
theorem iUnion_Ioc_eq_Ioi_self_iff {f : ι → α} {a : α} :
    ⋃ i, Ioc a (f i) = Ioi a ↔ ∀ x, a < x → ∃ i, x ≤ f i 
goal: ⋃ i, Set.Ioc a (f i) = Set.Ioi a ↔ ∀ (x : α), a < x → ∃ i, x ≤ f i

full_name: Set.biUnion_Ico_eq_Iio_self_iff
code: @[simp]
theorem biUnion_Ico_eq_Iio_self_iff {p : ι → Prop} {f : ∀ i, p i → α} {a : α} :
    ⋃ (i) (hi : p i), Ico (f i hi) a = Iio a ↔ ∀ x < a, ∃ i hi, f i hi ≤ x 
goal: ⋃ i, ⋃ (hi : p i), Set.Ico (f i hi) a = Set.Iio a ↔ ∀ x < a, ∃ i, ∃ (hi : p i), f i hi ≤ x

full_name: Set.biUnion_Ioc_eq_Ioi_self_iff
code: @[simp]
theorem biUnion_Ioc_eq_Ioi_self_iff {p : ι → Prop} {f : ∀ i, p i → α} {a : α} :
    ⋃ (i) (hi : p i), Ioc a (f i hi) = Ioi a ↔ ∀ x, a < x → ∃ i hi, x ≤ f i hi 
goal: ⋃ i, ⋃ (hi : p i), Set.Ioc a (f i hi) = Set.Ioi a ↔ ∀ (x : α), a < x → ∃ i, ∃ (hi : p i), x ≤ f i hi

full_name: IsGLB.biUnion_Ioi_eq
code: theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : ⋃ x ∈ s, Ioi x = Ioi a 
goal: ⋃ x ∈ s, Set.Ioi x = Set.Ioi a

full_name: IsGLB.iUnion_Ioi_eq
code: theorem IsGLB.iUnion_Ioi_eq (h : IsGLB (range f) a) : ⋃ x, Ioi (f x) = Ioi a 
goal: ⋃ x, Set.Ioi (f x) = Set.Ioi a

full_name: IsLUB.biUnion_Iio_eq
code: theorem IsLUB.biUnion_Iio_eq (h : IsLUB s a) : ⋃ x ∈ s, Iio x = Iio a 
goal: ⋃ x ∈ s, Set.Iio x = Set.Iio a

full_name: IsLUB.iUnion_Iio_eq
code: theorem IsLUB.iUnion_Iio_eq (h : IsLUB (range f) a) : ⋃ x, Iio (f x) = Iio a 
goal: ⋃ x, Set.Iio (f x) = Set.Iio a

full_name: IsGLB.biUnion_Ici_eq_Ioi
code: theorem IsGLB.biUnion_Ici_eq_Ioi (a_glb : IsGLB s a) (a_not_mem : a ∉ s) :
    ⋃ x ∈ s, Ici x = Ioi a 
goal: ⋃ x ∈ s, Set.Ici x = Set.Ioi a

full_name: IsGLB.biUnion_Ici_eq_Ici
code: theorem IsGLB.biUnion_Ici_eq_Ici (a_glb : IsGLB s a) (a_mem : a ∈ s) :
    ⋃ x ∈ s, Ici x = Ici a 
goal: ⋃ x ∈ s, Set.Ici x = Set.Ici a

full_name: IsLUB.biUnion_Iic_eq_Iio
code: theorem IsLUB.biUnion_Iic_eq_Iio (a_lub : IsLUB s a) (a_not_mem : a ∉ s) :
    ⋃ x ∈ s, Iic x = Iio a 
goal: ⋃ x ∈ s, Set.Iic x = Set.Iio a

full_name: IsLUB.biUnion_Iic_eq_Iic
code: theorem IsLUB.biUnion_Iic_eq_Iic (a_lub : IsLUB s a) (a_mem : a ∈ s) : ⋃ x ∈ s, Iic x = Iic a 
goal: ⋃ x ∈ s, Set.Iic x = Set.Iic a

full_name: iUnion_Ici_eq_Ioi_iInf
code: theorem iUnion_Ici_eq_Ioi_iInf {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (no_least_elem : ⨅ i, f i ∉ range f) : ⋃ i : ι, Ici (f i) = Ioi (⨅ i, f i) 
goal: ⋃ i, Set.Ici (f i) = Set.Ioi (⨅ i, f i)

full_name: iUnion_Iic_eq_Iio_iSup
code: theorem iUnion_Iic_eq_Iio_iSup {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (no_greatest_elem : (⨆ i, f i) ∉ range f) : ⋃ i : ι, Iic (f i) = Iio (⨆ i, f i) 
goal: ⋃ i, Set.Iic (f i) = Set.Iio (⨆ i, f i)

full_name: iUnion_Ici_eq_Ici_iInf
code: theorem iUnion_Ici_eq_Ici_iInf {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (has_least_elem : (⨅ i, f i) ∈ range f) : ⋃ i : ι, Ici (f i) = Ici (⨅ i, f i) 
goal: ⋃ i, Set.Ici (f i) = Set.Ici (⨅ i, f i)

full_name: iUnion_Iic_eq_Iic_iSup
code: theorem iUnion_Iic_eq_Iic_iSup {R : Type*} [CompleteLinearOrder R] {f : ι → R}
    (has_greatest_elem : (⨆ i, f i) ∈ range f) : ⋃ i : ι, Iic (f i) = Iic (⨆ i, f i) 
goal: ⋃ i, Set.Iic (f i) = Set.Iic (⨆ i, f i)

full_name: iUnion_Iio_eq_univ_iff
code: theorem iUnion_Iio_eq_univ_iff : ⋃ i, Iio (f i) = univ ↔ (¬ BddAbove (range f)) 
goal: ⋃ i, Set.Iio (f i) = Set.univ ↔ ¬BddAbove (Set.range f)

full_name: iUnion_Iic_of_not_bddAbove_range
code: theorem iUnion_Iic_of_not_bddAbove_range (hf : ¬ BddAbove (range f)) : ⋃ i, Iic (f i) = univ 
goal: ⋃ i, Set.Iic (f i) = Set.univ

full_name: iInter_Iic_eq_empty_iff
code: theorem iInter_Iic_eq_empty_iff : ⋂ i, Iic (f i) = ∅ ↔ ¬ BddBelow (range f) 
goal: ⋂ i, Set.Iic (f i) = ∅ ↔ ¬BddBelow (Set.range f)

full_name: iInter_Iio_of_not_bddBelow_range
code: theorem iInter_Iio_of_not_bddBelow_range (hf : ¬ BddBelow (range f)) : ⋂ i, Iio (f i) = ∅ 
goal: ⋂ i, Set.Iio (f i) = ∅