File: Mathlib/Data/Finset/Slice.lean

full_name: Set.Sized
code: /-- `Sized r A` means that every Finset in `A` has size `r`. -/
def Sized (r : ℕ) (A : Set (Finset α)) : Prop 
goal: Prop

full_name: Set.Sized.mono
code: theorem Sized.mono (h : A ⊆ B) (hB : B.Sized r) : A.Sized r 
goal: Set.Sized r A

full_name: Set.sized_empty
code: @[simp] lemma sized_empty : (∅ : Set (Finset α)).Sized r 
goal: Set.Sized r ∅

full_name: Set.sized_singleton
code: @[simp] lemma sized_singleton : ({s} : Set (Finset α)).Sized r ↔ s.card = r 
goal: Set.Sized r {s} ↔ s.card = r

full_name: Set.sized_union
code: theorem sized_union : (A ∪ B).Sized r ↔ A.Sized r ∧ B.Sized r 
goal: Set.Sized r (A ∪ B) ↔ Set.Sized r A ∧ Set.Sized r B

full_name: Set.sized_iUnion
code: @[simp]
theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r 
goal: Set.Sized r (⋃ i, f i) ↔ ∀ (i : ι), Set.Sized r (f i)

full_name: Set.sized_iUnion₂
code: theorem sized_iUnion₂ {f : ∀ i, κ i → Set (Finset α)} :
    (⋃ (i) (j), f i j).Sized r ↔ ∀ i j, (f i j).Sized r 
goal: Set.Sized r (⋃ i, ⋃ j, f i j) ↔ ∀ (i : ι) (j : κ i), Set.Sized r (f i j)

full_name: Set.Sized.isAntichain
code: protected theorem Sized.isAntichain (hA : A.Sized r) : IsAntichain (· ⊆ ·) A 
goal: IsAntichain (fun x x_1 => x ⊆ x_1) A

full_name: Set.Sized.subsingleton
code: protected theorem Sized.subsingleton (hA : A.Sized 0) : A.Subsingleton 
goal: A.Subsingleton

full_name: Set.Sized.subsingleton'
code: theorem Sized.subsingleton' [Fintype α] (hA : A.Sized (Fintype.card α)) : A.Subsingleton 
goal: A.Subsingleton

full_name: Set.Sized.empty_mem_iff
code: theorem Sized.empty_mem_iff (hA : A.Sized r) : ∅ ∈ A ↔ A = {∅} 
goal: ∅ ∈ A ↔ A = {∅}

full_name: Set.Sized.univ_mem_iff
code: theorem Sized.univ_mem_iff [Fintype α] (hA : A.Sized r) : Finset.univ ∈ A ↔ A = {Finset.univ} 
goal: Finset.univ ∈ A ↔ A = {Finset.univ}

full_name: Set.sized_powersetCard
code: theorem sized_powersetCard (s : Finset α) (r : ℕ) : (powersetCard r s : Set (Finset α)).Sized r 
goal: Set.Sized r ↑(Finset.powersetCard r s)

full_name: Finset.subset_powersetCard_univ_iff
code: theorem subset_powersetCard_univ_iff : 𝒜 ⊆ powersetCard r univ ↔ (𝒜 : Set (Finset α)).Sized r 
goal: 𝒜 ⊆ Finset.powersetCard r Finset.univ ↔ Set.Sized r ↑𝒜

full_name: Set.Sized.card_le
code: theorem _root_.Set.Sized.card_le (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
    card 𝒜 ≤ (Fintype.card α).choose r 
goal: 𝒜.card ≤ (Fintype.card α).choose r

full_name: Finset.slice
code: /-- The `r`-th slice of a set family is the subset of its elements which have cardinality `r`. -/
def slice (𝒜 : Finset (Finset α)) (r : ℕ) : Finset (Finset α) 
goal: Finset (Finset α)

full_name: Finset._aux_Mathlib_Data_Finset_Slice___unexpand_Finset_slice_1
code: @[inherit_doc]
scoped[Finset] infixl:90 " # " => Finset.slice
goal: Lean.PrettyPrinter.Unexpander

full_name: Finset.«_aux_Mathlib_Data_Finset_Slice___macroRules_Finset_term_#__1»
code: @[inherit_doc]
scoped[Finset] infixl:90 " # " => Finset.slice
goal: Lean.Macro

full_name: Finset.mem_slice
code: /-- `A` is in the `r`-th slice of `𝒜` iff it's in `𝒜` and has cardinality `r`. -/
theorem mem_slice : A ∈ 𝒜 # r ↔ A ∈ 𝒜 ∧ A.card = r 
goal: A ∈ 𝒜 # r ↔ A ∈ 𝒜 ∧ A.card = r

full_name: Finset.slice_subset
code: /-- The `r`-th slice of `𝒜` is a subset of `𝒜`. -/
theorem slice_subset : 𝒜 # r ⊆ 𝒜 
goal: 𝒜 # r ⊆ 𝒜

full_name: Finset.sized_slice
code: /-- Everything in the `r`-th slice of `𝒜` has size `r`. -/
theorem sized_slice : (𝒜 # r : Set (Finset α)).Sized r 
goal: Set.Sized r ↑(𝒜 # r)

full_name: Finset.eq_of_mem_slice
code: theorem eq_of_mem_slice (h₁ : A ∈ 𝒜 # r₁) (h₂ : A ∈ 𝒜 # r₂) : r₁ = r₂ 
goal: r₁ = r₂

full_name: Finset.ne_of_mem_slice
code: /-- Elements in distinct slices must be distinct. -/
theorem ne_of_mem_slice (h₁ : A₁ ∈ 𝒜 # r₁) (h₂ : A₂ ∈ 𝒜 # r₂) : r₁ ≠ r₂ → A₁ ≠ A₂ 
goal: r₁ ≠ r₂ → A₁ ≠ A₂

full_name: Finset.pairwiseDisjoint_slice
code: theorem pairwiseDisjoint_slice : (Set.univ : Set ℕ).PairwiseDisjoint (slice 𝒜) 
goal: Set.univ.PairwiseDisjoint 𝒜.slice

full_name: Finset.biUnion_slice
code: @[simp]
theorem biUnion_slice [DecidableEq α] : (Iic <| Fintype.card α).biUnion 𝒜.slice = 𝒜 
goal: (Finset.Iic (Fintype.card α)).biUnion 𝒜.slice = 𝒜

full_name: Finset.sum_card_slice
code: @[simp]
theorem sum_card_slice : (∑ r ∈ Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card 
goal: ∑ r ∈ Finset.Iic (Fintype.card α), (𝒜 # r).card = 𝒜.card