File: Mathlib/Topology/MetricSpace/MetricSeparated.lean

full_name: IsMetricSeparated
code: /-- Two sets in an (extended) metric space are called *metric separated* if the (extended) distance
between `x ∈ s` and `y ∈ t` is bounded from below by a positive constant. -/
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) 
goal: Prop

full_name: IsMetricSeparated.symm
code: @[symm]
theorem symm (h : IsMetricSeparated s t) : IsMetricSeparated t s 
goal: IsMetricSeparated t s

full_name: IsMetricSeparated.comm
code: theorem comm : IsMetricSeparated s t ↔ IsMetricSeparated t s 
goal: IsMetricSeparated s t ↔ IsMetricSeparated t s

full_name: IsMetricSeparated.empty_left
code: @[simp]
theorem empty_left (s : Set X) : IsMetricSeparated ∅ s 
goal: IsMetricSeparated ∅ s

full_name: IsMetricSeparated.empty_right
code: @[simp]
theorem empty_right (s : Set X) : IsMetricSeparated s ∅ 
goal: IsMetricSeparated s ∅

full_name: IsMetricSeparated.disjoint
code: protected theorem disjoint (h : IsMetricSeparated s t) : Disjoint s t 
goal: Disjoint s t

full_name: IsMetricSeparated.subset_compl_right
code: theorem subset_compl_right (h : IsMetricSeparated s t) : s ⊆ tᶜ 
goal: s ⊆ tᶜ

full_name: IsMetricSeparated.mono
code: @[mono]
theorem mono {s' t'} (hs : s ⊆ s') (ht : t ⊆ t') :
    IsMetricSeparated s' t' → IsMetricSeparated s t 
goal: IsMetricSeparated s' t' → IsMetricSeparated s t

full_name: IsMetricSeparated.mono_left
code: theorem mono_left {s'} (h' : IsMetricSeparated s' t) (hs : s ⊆ s') : IsMetricSeparated s t 
goal: IsMetricSeparated s t

full_name: IsMetricSeparated.mono_right
code: theorem mono_right {t'} (h' : IsMetricSeparated s t') (ht : t ⊆ t') : IsMetricSeparated s t 
goal: IsMetricSeparated s t

full_name: IsMetricSeparated.union_left
code: theorem union_left {s'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s' t) :
    IsMetricSeparated (s ∪ s') t 
goal: IsMetricSeparated (s ∪ s') t

full_name: IsMetricSeparated.union_left_iff
code: @[simp]
theorem union_left_iff {s'} :
    IsMetricSeparated (s ∪ s') t ↔ IsMetricSeparated s t ∧ IsMetricSeparated s' t 
goal: IsMetricSeparated (s ∪ s') t ↔ IsMetricSeparated s t ∧ IsMetricSeparated s' t

full_name: IsMetricSeparated.union_right
code: theorem union_right {t'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s t') :
    IsMetricSeparated s (t ∪ t') 
goal: IsMetricSeparated s (t ∪ t')

full_name: IsMetricSeparated.union_right_iff
code: @[simp]
theorem union_right_iff {t'} :
    IsMetricSeparated s (t ∪ t') ↔ IsMetricSeparated s t ∧ IsMetricSeparated s t' 
goal: IsMetricSeparated s (t ∪ t') ↔ IsMetricSeparated s t ∧ IsMetricSeparated s t'

full_name: IsMetricSeparated.finite_iUnion_left_iff
code: theorem finite_iUnion_left_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set X}
    {t : Set X} : IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t 
goal: IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t

full_name: IsMetricSeparated.finite_iUnion_right_iff
code: theorem finite_iUnion_right_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : Set X}
    {t : ι → Set X} : IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) 
goal: IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i)

full_name: IsMetricSeparated.finset_iUnion_left_iff
code: @[simp]
theorem finset_iUnion_left_iff {ι : Type*} {I : Finset ι} {s : ι → Set X} {t : Set X} :
    IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t 
goal: IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t

full_name: IsMetricSeparated.finset_iUnion_right_iff
code: @[simp]
theorem finset_iUnion_right_iff {ι : Type*} {I : Finset ι} {s : Set X} {t : ι → Set X} :
    IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) 
goal: IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i)