File: Mathlib/Combinatorics/SetFamily/Intersecting.lean

full_name: Set.Intersecting
code: /-- A set family is intersecting if every pair of elements is non-disjoint. -/
def Intersecting (s : Set α) : Prop 
goal: Prop

full_name: Set.Intersecting.mono
code: @[mono]
theorem Intersecting.mono (h : t ⊆ s) (hs : s.Intersecting) : t.Intersecting 
goal: t.Intersecting

full_name: Set.Intersecting.not_bot_mem
code: theorem Intersecting.not_bot_mem (hs : s.Intersecting) : ⊥ ∉ s 
goal: ⊥ ∉ s

full_name: Set.Intersecting.ne_bot
code: theorem Intersecting.ne_bot (hs : s.Intersecting) (ha : a ∈ s) : a ≠ ⊥ 
goal: a ≠ ⊥

full_name: Set.intersecting_empty
code: theorem intersecting_empty : (∅ : Set α).Intersecting 
goal: ∅.Intersecting

full_name: Set.intersecting_singleton
code: @[simp]
theorem intersecting_singleton : ({a} : Set α).Intersecting ↔ a ≠ ⊥ 
goal: {a}.Intersecting ↔ a ≠ ⊥

full_name: Set.Intersecting.insert
code: protected theorem Intersecting.insert (hs : s.Intersecting) (ha : a ≠ ⊥)
    (h : ∀ b ∈ s, ¬Disjoint a b) : (insert a s).Intersecting 
goal: (insert a s).Intersecting

full_name: Set.intersecting_insert
code: theorem intersecting_insert :
    (insert a s).Intersecting ↔ s.Intersecting ∧ a ≠ ⊥ ∧ ∀ b ∈ s, ¬Disjoint a b 
goal: (insert a s).Intersecting ↔ s.Intersecting ∧ a ≠ ⊥ ∧ ∀ b ∈ s, ¬Disjoint a b

full_name: Set.intersecting_iff_pairwise_not_disjoint
code: theorem intersecting_iff_pairwise_not_disjoint :
    s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥} 
goal: s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥}

full_name: Set.Subsingleton.intersecting
code: protected theorem Subsingleton.intersecting (hs : s.Subsingleton) : s.Intersecting ↔ s ≠ {⊥} 
goal: s.Intersecting ↔ s ≠ {⊥}

full_name: Set.intersecting_iff_eq_empty_of_subsingleton
code: theorem intersecting_iff_eq_empty_of_subsingleton [Subsingleton α] (s : Set α) :
    s.Intersecting ↔ s = ∅ 
goal: s.Intersecting ↔ s = ∅

full_name: Set.Intersecting.isUpperSet
code: /-- Maximal intersecting families are upper sets. -/
protected theorem Intersecting.isUpperSet (hs : s.Intersecting)
    (h : ∀ t : Set α, t.Intersecting → s ⊆ t → s = t) : IsUpperSet s 
goal: IsUpperSet s

full_name: Set.Intersecting.isUpperSet'
code: /-- Maximal intersecting families are upper sets. Finset version. -/
theorem Intersecting.isUpperSet' {s : Finset α} (hs : (s : Set α).Intersecting)
    (h : ∀ t : Finset α, (t : Set α).Intersecting → s ⊆ t → s = t) : IsUpperSet (s : Set α) 
goal: IsUpperSet ↑s

full_name: Set.Intersecting.exists_mem_set
code: theorem Intersecting.exists_mem_set {𝒜 : Set (Set α)} (h𝒜 : 𝒜.Intersecting) {s t : Set α}
    (hs : s ∈ 𝒜) (ht : t ∈ 𝒜) : ∃ a, a ∈ s ∧ a ∈ t 
goal: ∃ a ∈ s, a ∈ t

full_name: Set.Intersecting.exists_mem_finset
code: theorem Intersecting.exists_mem_finset [DecidableEq α] {𝒜 : Set (Finset α)} (h𝒜 : 𝒜.Intersecting)
    {s t : Finset α} (hs : s ∈ 𝒜) (ht : t ∈ 𝒜) : ∃ a, a ∈ s ∧ a ∈ t 
goal: ∃ a ∈ s, a ∈ t

full_name: Set.Intersecting.not_compl_mem
code: theorem Intersecting.not_compl_mem {s : Set α} (hs : s.Intersecting) {a : α} (ha : a ∈ s) :
    aᶜ ∉ s 
goal: aᶜ ∉ s

full_name: Set.Intersecting.not_mem
code: theorem Intersecting.not_mem {s : Set α} (hs : s.Intersecting) {a : α} (ha : aᶜ ∈ s) : a ∉ s 
goal: a ∉ s

full_name: Set.Intersecting.disjoint_map_compl
code: theorem Intersecting.disjoint_map_compl {s : Finset α} (hs : (s : Set α).Intersecting) :
    Disjoint s (s.map ⟨compl, compl_injective⟩) 
goal: Disjoint s (Finset.map { toFun := compl, inj' := ⋯ } s)

full_name: Set.Intersecting.card_le
code: theorem Intersecting.card_le [Fintype α] {s : Finset α} (hs : (s : Set α).Intersecting) :
    2 * s.card ≤ Fintype.card α 
goal: 2 * s.card ≤ Fintype.card α

full_name: Set.Intersecting.is_max_iff_card_eq
code: theorem Intersecting.is_max_iff_card_eq (hs : (s : Set α).Intersecting) :
    (∀ t : Finset α, (t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * s.card = Fintype.card α 
goal: (∀ (t : Finset α), (↑t).Intersecting → s ⊆ t → s = t) ↔ 2 * s.card = Fintype.card α

full_name: Set.Intersecting.exists_card_eq
code: theorem Intersecting.exists_card_eq (hs : (s : Set α).Intersecting) :
    ∃ t, s ⊆ t ∧ 2 * t.card = Fintype.card α ∧ (t : Set α).Intersecting 
goal: ∃ t, s ⊆ t ∧ 2 * t.card = Fintype.card α ∧ (↑t).Intersecting