Documentation

Mathlib.Data.Finset.Erase

Erasing an element from a finite set #

Main declarations #

Tags #

finite sets, finset

erase #

source
def Finset.erase {α : Type u_1} [DecidableEq α] (s : Finset α) (a : α) :
Finset α

erase s a is the set s - {a}, that is, the elements of s which are not equal to a.

Equations
    Instances For
      source
      @[simp]
      theorem Finset.erase_val {α : Type u_1} [DecidableEq α] (s : Finset α) (a : α) :
      (s.erase a).val = s.val.erase a
      source
      @[simp]
      theorem Finset.mem_erase {α : Type u_1} [DecidableEq α] {a b : α} {s : Finset α} :
      a ∈ s.erase b ↔ a ≠ b ∧ a ∈ s
      source
      theorem Finset.notMem_erase {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
      a ∉ s.erase a
      source
      @[deprecated Finset.notMem_erase (since := "2025-05-23")]
      theorem Finset.not_mem_erase {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
      a ∉ s.erase a

      Alias of Finset.notMem_erase.

      source
      theorem Finset.ne_of_mem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} :
      b ∈ s.erase a → b ≠ a
      source
      theorem Finset.mem_of_mem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} :
      b ∈ s.erase a → b ∈ s
      source
      theorem Finset.mem_erase_of_ne_of_mem {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} :
      a ≠ b → a ∈ s → a ∈ s.erase b
      source
      theorem Finset.eq_of_mem_of_notMem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} (hs : b ∈ s) (hsa : b ∉ s.erase a) :
      b = a

      An element of s that is not an element of erase s a must bea.

      source
      @[deprecated Finset.eq_of_mem_of_notMem_erase (since := "2025-05-23")]
      theorem Finset.eq_of_mem_of_not_mem_erase {α : Type u_1} [DecidableEq α] {s : Finset α} {a b : α} (hs : b ∈ s) (hsa : b ∉ s.erase a) :
      b = a

      Alias of Finset.eq_of_mem_of_notMem_erase.

      An element of s that is not an element of erase s a must bea.

      source
      @[simp]
      theorem Finset.erase_eq_of_notMem {α : Type u_1} [DecidableEq α] {a : α} {s : Finset α} (h : a ∉ s) :
      s.erase a = s
      source
      @[deprecated Finset.erase_eq_of_notMem (since := "2025-05-23")]
      theorem Finset.erase_eq_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Finset α} (h : a ∉ s) :
      s.erase a = s

      Alias of Finset.erase_eq_of_notMem.

      source
      @[simp]
      theorem Finset.erase_eq_self {α : Type u_1} [DecidableEq α] {s : Finset α} {a : α} :
      s.erase a = s ↔ a ∉ s
      source
      theorem Finset.erase_ne_self {α : Type u_1} [DecidableEq α] {s : Finset α} {a : α} :
      s.erase a ≠ s ↔ a ∈ s
      source
      theorem Finset.erase_subset_erase {α : Type u_1} [DecidableEq α] (a : α) {s t : Finset α} (h : s ⊆ t) :
      s.erase a ⊆ t.erase a
      source
      theorem Finset.erase_subset {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
      s.erase a ⊆ s
      source
      theorem Finset.subset_erase {α : Type u_1} [DecidableEq α] {a : α} {s t : Finset α} :
      s ⊆ t.erase a ↔ s ⊆ t ∧ a ∉ s
      source
      @[simp]
      theorem Finset.coe_erase {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) :
      ↑(s.erase a) = ↑s \ {a}
      source
      theorem Finset.erase_idem {α : Type u_1} [DecidableEq α] {a : α} {s : Finset α} :
      (s.erase a).erase a = s.erase a
      source
      theorem Finset.erase_right_comm {α : Type u_1} [DecidableEq α] {a b : α} {s : Finset α} :
      (s.erase a).erase b = (s.erase b).erase a
      source
      theorem Finset.erase_inj {α : Type u_1} [DecidableEq α] {x y : α} (s : Finset α) (hx : x ∈ s) :
      s.erase x = s.erase y ↔ x = y
      source
      theorem Finset.erase_injOn {α : Type u_1} [DecidableEq α] (s : Finset α) :
      Set.InjOn s.erase ↑s