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

full_name: Set.ordConnectedComponent
code: /-- Order-connected component of a point `x` in a set `s`. It is defined as the set of `y` such that
`Set.uIcc x y ⊆ s`. Note that it is empty if and only if `x ∉ s`. -/
def ordConnectedComponent (s : Set α) (x : α) : Set α 
goal: Set α

full_name: Set.mem_ordConnectedComponent
code: theorem mem_ordConnectedComponent : y ∈ ordConnectedComponent s x ↔ [[x, y]] ⊆ s 
goal: y ∈ s.ordConnectedComponent x ↔ [[x, y]] ⊆ s

full_name: Set.dual_ordConnectedComponent
code: theorem dual_ordConnectedComponent :
    ordConnectedComponent (ofDual ⁻¹' s) (toDual x) = ofDual ⁻¹' ordConnectedComponent s x 
goal: (⇑OrderDual.ofDual ⁻¹' s).ordConnectedComponent (OrderDual.toDual x) = ⇑OrderDual.ofDual ⁻¹' s.ordConnectedComponent x

full_name: Set.ordConnectedComponent_subset
code: theorem ordConnectedComponent_subset : ordConnectedComponent s x ⊆ s 
goal: s.ordConnectedComponent x ⊆ s

full_name: Set.subset_ordConnectedComponent
code: theorem subset_ordConnectedComponent {t} [h : OrdConnected s] (hs : x ∈ s) (ht : s ⊆ t) :
    s ⊆ ordConnectedComponent t x 
goal: s ⊆ t.ordConnectedComponent x

full_name: Set.self_mem_ordConnectedComponent
code: @[simp]
theorem self_mem_ordConnectedComponent : x ∈ ordConnectedComponent s x ↔ x ∈ s 
goal: x ∈ s.ordConnectedComponent x ↔ x ∈ s

full_name: Set.nonempty_ordConnectedComponent
code: @[simp]
theorem nonempty_ordConnectedComponent : (ordConnectedComponent s x).Nonempty ↔ x ∈ s 
goal: (s.ordConnectedComponent x).Nonempty ↔ x ∈ s

full_name: Set.ordConnectedComponent_eq_empty
code: @[simp]
theorem ordConnectedComponent_eq_empty : ordConnectedComponent s x = ∅ ↔ x ∉ s 
goal: s.ordConnectedComponent x = ∅ ↔ x ∉ s

full_name: Set.ordConnectedComponent_empty
code: @[simp]
theorem ordConnectedComponent_empty : ordConnectedComponent ∅ x = ∅ 
goal: ∅.ordConnectedComponent x = ∅

full_name: Set.ordConnectedComponent_univ
code: @[simp]
theorem ordConnectedComponent_univ : ordConnectedComponent univ x = univ 
goal: Set.univ.ordConnectedComponent x = Set.univ

full_name: Set.ordConnectedComponent_inter
code: theorem ordConnectedComponent_inter (s t : Set α) (x : α) :
    ordConnectedComponent (s ∩ t) x = ordConnectedComponent s x ∩ ordConnectedComponent t x 
goal: (s ∩ t).ordConnectedComponent x = s.ordConnectedComponent x ∩ t.ordConnectedComponent x

full_name: Set.mem_ordConnectedComponent_comm
code: theorem mem_ordConnectedComponent_comm :
    y ∈ ordConnectedComponent s x ↔ x ∈ ordConnectedComponent s y 
goal: y ∈ s.ordConnectedComponent x ↔ x ∈ s.ordConnectedComponent y

full_name: Set.mem_ordConnectedComponent_trans
code: theorem mem_ordConnectedComponent_trans (hxy : y ∈ ordConnectedComponent s x)
    (hyz : z ∈ ordConnectedComponent s y) : z ∈ ordConnectedComponent s x 
goal: z ∈ s.ordConnectedComponent x

full_name: Set.ordConnectedComponent_eq
code: theorem ordConnectedComponent_eq (h : [[x, y]] ⊆ s) :
    ordConnectedComponent s x = ordConnectedComponent s y 
goal: s.ordConnectedComponent x = s.ordConnectedComponent y

full_name: Set.instOrdConnectedOrdConnectedComponent
code: instance : OrdConnected (ordConnectedComponent s x) 
goal: (s.ordConnectedComponent x).OrdConnected

full_name: Set.ordConnectedProj
code: /-- Projection from `s : Set α` to `α` sending each order connected component of `s` to a single
point of this component. -/
noncomputable def ordConnectedProj (s : Set α) : s → α 
goal: ↑s → α

full_name: Set.ordConnectedProj_mem_ordConnectedComponent
code: theorem ordConnectedProj_mem_ordConnectedComponent (s : Set α) (x : s) :
    ordConnectedProj s x ∈ ordConnectedComponent s x 
goal: s.ordConnectedProj x ∈ s.ordConnectedComponent ↑x

full_name: Set.mem_ordConnectedComponent_ordConnectedProj
code: theorem mem_ordConnectedComponent_ordConnectedProj (s : Set α) (x : s) :
    ↑x ∈ ordConnectedComponent s (ordConnectedProj s x) 
goal: ↑x ∈ s.ordConnectedComponent (s.ordConnectedProj x)

full_name: Set.ordConnectedComponent_ordConnectedProj
code: @[simp]
theorem ordConnectedComponent_ordConnectedProj (s : Set α) (x : s) :
    ordConnectedComponent s (ordConnectedProj s x) = ordConnectedComponent s x 
goal: s.ordConnectedComponent (s.ordConnectedProj x) = s.ordConnectedComponent ↑x

full_name: Set.ordConnectedProj_eq
code: @[simp]
theorem ordConnectedProj_eq {x y : s} :
    ordConnectedProj s x = ordConnectedProj s y ↔ [[(x : α), y]] ⊆ s 
goal: s.ordConnectedProj x = s.ordConnectedProj y ↔ [[↑x, ↑y]] ⊆ s

full_name: Set.ordConnectedSection
code: /-- A set that intersects each order connected component of a set by a single point. Defined as the
range of `Set.ordConnectedProj s`. -/
def ordConnectedSection (s : Set α) : Set α 
goal: Set α

full_name: Set.dual_ordConnectedSection
code: theorem dual_ordConnectedSection (s : Set α) :
    ordConnectedSection (ofDual ⁻¹' s) = ofDual ⁻¹' ordConnectedSection s 
goal: (⇑OrderDual.ofDual ⁻¹' s).ordConnectedSection = ⇑OrderDual.ofDual ⁻¹' s.ordConnectedSection

full_name: Set.ordConnectedSection_subset
code: theorem ordConnectedSection_subset : ordConnectedSection s ⊆ s 
goal: s.ordConnectedSection ⊆ s

full_name: Set.eq_of_mem_ordConnectedSection_of_uIcc_subset
code: theorem eq_of_mem_ordConnectedSection_of_uIcc_subset (hx : x ∈ ordConnectedSection s)
    (hy : y ∈ ordConnectedSection s) (h : [[x, y]] ⊆ s) : x = y 
goal: x = y

full_name: Set.ordSeparatingSet
code: /-- Given two sets `s t : Set α`, the set `Set.orderSeparatingSet s t` is the set of points that
belong both to some `Set.ordConnectedComponent tᶜ x`, `x ∈ s`, and to some
`Set.ordConnectedComponent sᶜ x`, `x ∈ t`. In the case of two disjoint closed sets, this is the
union of all open intervals $(a, b)$ such that their endpoints belong to different sets. -/
def ordSeparatingSet (s t : Set α) : Set α 
goal: Set α

full_name: Set.ordSeparatingSet_comm
code: theorem ordSeparatingSet_comm (s t : Set α) : ordSeparatingSet s t = ordSeparatingSet t s 
goal: s.ordSeparatingSet t = t.ordSeparatingSet s

full_name: Set.disjoint_left_ordSeparatingSet
code: theorem disjoint_left_ordSeparatingSet : Disjoint s (ordSeparatingSet s t) 
goal: Disjoint s (s.ordSeparatingSet t)

full_name: Set.disjoint_right_ordSeparatingSet
code: theorem disjoint_right_ordSeparatingSet : Disjoint t (ordSeparatingSet s t) 
goal: Disjoint t (s.ordSeparatingSet t)

full_name: Set.dual_ordSeparatingSet
code: theorem dual_ordSeparatingSet :
    ordSeparatingSet (ofDual ⁻¹' s) (ofDual ⁻¹' t) = ofDual ⁻¹' ordSeparatingSet s t 
goal: (⇑OrderDual.ofDual ⁻¹' s).ordSeparatingSet (⇑OrderDual.ofDual ⁻¹' t) = ⇑OrderDual.ofDual ⁻¹' s.ordSeparatingSet t

full_name: Set.ordT5Nhd
code: /-- An auxiliary neighborhood that will be used in the proof of
    `OrderTopology.CompletelyNormalSpace`. -/
def ordT5Nhd (s t : Set α) : Set α 
goal: Set α

full_name: Set.disjoint_ordT5Nhd
code: theorem disjoint_ordT5Nhd : Disjoint (ordT5Nhd s t) (ordT5Nhd t s) 
goal: Disjoint (s.ordT5Nhd t) (t.ordT5Nhd s)