ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.Nonempty.eq_zero
theorem Nonempty.eq_zero [Subsingleton α] [Zero α] {s : Set α} (h : s.Nonempty) :
    s = {0} 
Arguments: {α : Type u} [Subsingleton α] [Zero α] {s : Set α} (h : s.Nonempty)
Goal: s = {0}
-- Set.Nonempty.eq_one
theorem Nonempty.eq_one [Subsingleton α] [One α] {s : Set α} (h : s.Nonempty) :
    s = {1} 
Arguments: {α : Type u} [Subsingleton α] [One α] {s : Set α} (h : s.Nonempty)
Goal: s = {1}
-- Set.disjoint_iff
protected theorem disjoint_iff : Disjoint s t ↔ s ∩ t ⊆ ∅ 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s t ↔ s ∩ t ⊆ ∅
-- Set.disjoint_iff_inter_eq_empty
theorem disjoint_iff_inter_eq_empty : Disjoint s t ↔ s ∩ t = ∅ 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s t ↔ s ∩ t = ∅
-- Disjoint.inter_eq
theorem _root_.Disjoint.inter_eq : Disjoint s t → s ∩ t = ∅ 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s t → s ∩ t = ∅
-- Set.disjoint_left
theorem disjoint_left : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → a ∉ t 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s t ↔ ∀ ⦃a : α⦄, a ∈ s → a ∉ t
-- Set.disjoint_right
theorem disjoint_right : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ t → a ∉ s 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s t ↔ ∀ ⦃a : α⦄, a ∈ t → a ∉ s
-- Set.not_disjoint_iff
lemma not_disjoint_iff : ¬Disjoint s t ↔ ∃ x, x ∈ s ∧ x ∈ t 
Arguments: {α : Type u} {s t : Set α}
Goal: ¬Disjoint s t ↔ ∃ x ∈ s, x ∈ t
-- Set.not_disjoint_iff_nonempty_inter
lemma not_disjoint_iff_nonempty_inter : ¬ Disjoint s t ↔ (s ∩ t).Nonempty 
Arguments: {α : Type u} {s t : Set α}
Goal: ¬Disjoint s t ↔ (s ∩ t).Nonempty
-- Set.disjoint_or_nonempty_inter
lemma disjoint_or_nonempty_inter (s t : Set α) : Disjoint s t ∨ (s ∩ t).Nonempty 
Arguments: {α : Type u} (s t : Set α)
Goal: Disjoint s t ∨ (s ∩ t).Nonempty
-- Set.disjoint_iff_forall_ne
lemma disjoint_iff_forall_ne : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ t → a ≠ b 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s t ↔ ∀ ⦃a : α⦄, a ∈ s → ∀ ⦃b : α⦄, b ∈ t → a ≠ b
-- Set.disjoint_of_subset_left
lemma disjoint_of_subset_left (h : s ⊆ u) (d : Disjoint u t) : Disjoint s t 
Arguments: {α : Type u} {s t u : Set α} (h : s ⊆ u) (d : Disjoint u t)
Goal: Disjoint s t
-- Set.disjoint_of_subset_right
lemma disjoint_of_subset_right (h : t ⊆ u) (d : Disjoint s u) : Disjoint s t 
Arguments: {α : Type u} {s t u : Set α} (h : t ⊆ u) (d : Disjoint s u)
Goal: Disjoint s t
-- Set.disjoint_of_subset
lemma disjoint_of_subset (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (h : Disjoint s₂ t₂) : Disjoint s₁ t₁ 
Arguments: {α : Type u} {s₁ s₂ t₁ t₂ : Set α} (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (h : Disjoint s₂ t₂)
Goal: Disjoint s₁ t₁
-- Set.disjoint_union_left
@[simp]
lemma disjoint_union_left : Disjoint (s ∪ t) u ↔ Disjoint s u ∧ Disjoint t u 
Arguments: {α : Type u} {s t u : Set α}
Goal: Disjoint (s ∪ t) u ↔ Disjoint s u ∧ Disjoint t u
-- Set.disjoint_union_right
@[simp]
lemma disjoint_union_right : Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u 
Arguments: {α : Type u} {s t u : Set α}
Goal: Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u
-- Set.disjoint_empty
@[simp] lemma disjoint_empty (s : Set α) : Disjoint s ∅ 
Arguments: {α : Type u} (s : Set α)
Goal: Disjoint s ∅
-- Set.empty_disjoint
@[simp] lemma empty_disjoint (s : Set α) : Disjoint ∅ s 
Arguments: {α : Type u} (s : Set α)
Goal: Disjoint ∅ s
-- Set.univ_disjoint
@[simp] lemma univ_disjoint : Disjoint univ s ↔ s = ∅ 
Arguments: {α : Type u} {s : Set α}
Goal: Disjoint Set.univ s ↔ s = ∅
-- Set.disjoint_univ
@[simp] lemma disjoint_univ : Disjoint s univ ↔ s = ∅ 
Arguments: {α : Type u} {s : Set α}
Goal: Disjoint s Set.univ ↔ s = ∅
-- Set.disjoint_sdiff_left
lemma disjoint_sdiff_left : Disjoint (t \ s) s 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint (t \ s) s
-- Set.disjoint_sdiff_right
lemma disjoint_sdiff_right : Disjoint s (t \ s) 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s (t \ s)
-- Set.disjoint_sdiff_inter
theorem disjoint_sdiff_inter : Disjoint (s \ t) (s ∩ t) 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint (s \ t) (s ∩ t)
-- Set.diff_union_diff_cancel
theorem diff_union_diff_cancel (hts : t ⊆ s) (hut : u ⊆ t) : s \ t ∪ t \ u = s \ u 
Arguments: {α : Type u} {s t u : Set α} (hts : t ⊆ s) (hut : u ⊆ t)
Goal: s \ t ∪ t \ u = s \ u
-- Set.diff_diff_eq_sdiff_union
theorem diff_diff_eq_sdiff_union (h : u ⊆ s) : s \ (t \ u) = s \ t ∪ u 
Arguments: {α : Type u} {s t u : Set α} (h : u ⊆ s)
Goal: s \ (t \ u) = s \ t ∪ u
-- Set.disjoint_singleton_left
@[simp default+1]
lemma disjoint_singleton_left : Disjoint {a} s ↔ a ∉ s 
Arguments: {α : Type u} {a : α} {s : Set α}
Goal: Disjoint {a} s ↔ a ∉ s
-- Set.disjoint_singleton_right
@[simp]
lemma disjoint_singleton_right : Disjoint s {a} ↔ a ∉ s 
Arguments: {α : Type u} {a : α} {s : Set α}
Goal: Disjoint s {a} ↔ a ∉ s
-- Set.disjoint_singleton
lemma disjoint_singleton : Disjoint ({a} : Set α) {b} ↔ a ≠ b 
Arguments: {α : Type u} {a b : α}
Goal: Disjoint {a} {b} ↔ a ≠ b
-- Set.subset_diff
lemma subset_diff : s ⊆ t \ u ↔ s ⊆ t ∧ Disjoint s u 
Arguments: {α : Type u} {s t u : Set α}
Goal: s ⊆ t \ u ↔ s ⊆ t ∧ Disjoint s u
-- Set.ssubset_iff_sdiff_singleton
lemma ssubset_iff_sdiff_singleton : s ⊂ t ↔ ∃ a ∈ t, s ⊆ t \ {a} 
Arguments: {α : Type u} {s t : Set α}
Goal: s ⊂ t ↔ ∃ a ∈ t, s ⊆ t \ {a}
-- Set.inter_diff_distrib_left
theorem inter_diff_distrib_left (s t u : Set α) : s ∩ (t \ u) = (s ∩ t) \ (s ∩ u) 
Arguments: {α : Type u} (s t u : Set α)
Goal: s ∩ (t \ u) = (s ∩ t) \ (s ∩ u)
-- Set.inter_diff_distrib_right
theorem inter_diff_distrib_right (s t u : Set α) : s \ t ∩ u = (s ∩ u) \ (t ∩ u) 
Arguments: {α : Type u} (s t u : Set α)
Goal: s \ t ∩ u = (s ∩ u) \ (t ∩ u)
-- Set.compl_def
theorem compl_def (s : Set α) : sᶜ = { x | x ∉ s } 
Arguments: {α : Type u} (s : Set α)
Goal: sᶜ = {x | x ∉ s}
-- Set.mem_compl
theorem mem_compl {s : Set α} {x : α} (h : x ∉ s) : x ∈ sᶜ 
Arguments: {α : Type u} {s : Set α} {x : α} (h : x ∉ s)
Goal: x ∈ sᶜ
-- Set.compl_setOf
theorem compl_setOf {α} (p : α → Prop) : { a | p a }ᶜ = { a | ¬p a } 
Arguments: {α : Type u_1} (p : α → Prop)
Goal: {a | p a}ᶜ = {a | ¬p a}
-- Set.not_mem_of_mem_compl
theorem not_mem_of_mem_compl {s : Set α} {x : α} (h : x ∈ sᶜ) : x ∉ s 
Arguments: {α : Type u} {s : Set α} {x : α} (h : x ∈ sᶜ)
Goal: x ∉ s
-- Set.not_mem_compl_iff
theorem not_mem_compl_iff {x : α} : x ∉ sᶜ ↔ x ∈ s 
Arguments: {α : Type u} {s : Set α} {x : α}
Goal: x ∉ sᶜ ↔ x ∈ s
-- Set.inter_compl_self
@[simp]
theorem inter_compl_self (s : Set α) : s ∩ sᶜ = ∅ 
Arguments: {α : Type u} (s : Set α)
Goal: s ∩ sᶜ = ∅
-- Set.compl_inter_self
@[simp]
theorem compl_inter_self (s : Set α) : sᶜ ∩ s = ∅ 
Arguments: {α : Type u} (s : Set α)
Goal: sᶜ ∩ s = ∅
-- Set.compl_empty
@[simp]
theorem compl_empty : (∅ : Set α)ᶜ = univ 
Arguments: {α : Type u}
Goal: ∅ᶜ = Set.univ
-- Set.compl_union
@[simp]
theorem compl_union (s t : Set α) : (s ∪ t)ᶜ = sᶜ ∩ tᶜ 
Arguments: {α : Type u} (s t : Set α)
Goal: (s ∪ t)ᶜ = sᶜ ∩ tᶜ
-- Set.compl_inter
theorem compl_inter (s t : Set α) : (s ∩ t)ᶜ = sᶜ ∪ tᶜ 
Arguments: {α : Type u} (s t : Set α)
Goal: (s ∩ t)ᶜ = sᶜ ∪ tᶜ
-- Set.compl_univ
@[simp]
theorem compl_univ : (univ : Set α)ᶜ = ∅ 
Arguments: {α : Type u}
Goal: Set.univᶜ = ∅
-- Set.compl_empty_iff
@[simp]
theorem compl_empty_iff {s : Set α} : sᶜ = ∅ ↔ s = univ 
Arguments: {α : Type u} {s : Set α}
Goal: sᶜ = ∅ ↔ s = Set.univ
-- Set.compl_univ_iff
@[simp]
theorem compl_univ_iff {s : Set α} : sᶜ = univ ↔ s = ∅ 
Arguments: {α : Type u} {s : Set α}
Goal: sᶜ = Set.univ ↔ s = ∅
-- Set.compl_ne_univ
theorem compl_ne_univ : sᶜ ≠ univ ↔ s.Nonempty 
Arguments: {α : Type u} {s : Set α}
Goal: sᶜ ≠ Set.univ ↔ s.Nonempty
-- Set.nonempty_compl
theorem nonempty_compl : sᶜ.Nonempty ↔ s ≠ univ 
Arguments: {α : Type u} {s : Set α}
Goal: sᶜ.Nonempty ↔ s ≠ Set.univ
-- Set.nonempty_compl_of_nontrivial
@[simp] lemma nonempty_compl_of_nontrivial [Nontrivial α] (x : α) : Set.Nonempty {x}ᶜ 
Arguments: {α : Type u} [Nontrivial α] (x : α)
Goal: {x}ᶜ.Nonempty
-- Set.mem_compl_singleton_iff
theorem mem_compl_singleton_iff {a x : α} : x ∈ ({a} : Set α)ᶜ ↔ x ≠ a 
Arguments: {α : Type u} {a x : α}
Goal: x ∈ {a}ᶜ ↔ x ≠ a
-- Set.compl_singleton_eq
theorem compl_singleton_eq (a : α) : ({a} : Set α)ᶜ = { x | x ≠ a } 
Arguments: {α : Type u} (a : α)
Goal: {a}ᶜ = {x | x ≠ a}
-- Set.compl_ne_eq_singleton
@[simp]
theorem compl_ne_eq_singleton (a : α) : ({ x | x ≠ a } : Set α)ᶜ = {a} 
Arguments: {α : Type u} (a : α)
Goal: {x | x ≠ a}ᶜ = {a}
-- Set.union_eq_compl_compl_inter_compl
theorem union_eq_compl_compl_inter_compl (s t : Set α) : s ∪ t = (sᶜ ∩ tᶜ)ᶜ 
Arguments: {α : Type u} (s t : Set α)
Goal: s ∪ t = (sᶜ ∩ tᶜ)ᶜ
-- Set.inter_eq_compl_compl_union_compl
theorem inter_eq_compl_compl_union_compl (s t : Set α) : s ∩ t = (sᶜ ∪ tᶜ)ᶜ 
Arguments: {α : Type u} (s t : Set α)
Goal: s ∩ t = (sᶜ ∪ tᶜ)ᶜ
-- Set.union_compl_self
@[simp]
theorem union_compl_self (s : Set α) : s ∪ sᶜ = univ 
Arguments: {α : Type u} (s : Set α)
Goal: s ∪ sᶜ = Set.univ
-- Set.compl_union_self
@[simp]
theorem compl_union_self (s : Set α) : sᶜ ∪ s = univ 
Arguments: {α : Type u} (s : Set α)
Goal: sᶜ ∪ s = Set.univ
-- Set.compl_subset_comm
theorem compl_subset_comm : sᶜ ⊆ t ↔ tᶜ ⊆ s 
Arguments: {α : Type u} {s t : Set α}
Goal: sᶜ ⊆ t ↔ tᶜ ⊆ s
-- Set.subset_compl_comm
theorem subset_compl_comm : s ⊆ tᶜ ↔ t ⊆ sᶜ 
Arguments: {α : Type u} {s t : Set α}
Goal: s ⊆ tᶜ ↔ t ⊆ sᶜ
-- Set.compl_subset_compl
@[simp]
theorem compl_subset_compl : sᶜ ⊆ tᶜ ↔ t ⊆ s 
Arguments: {α : Type u} {s t : Set α}
Goal: sᶜ ⊆ tᶜ ↔ t ⊆ s
-- Set.compl_subset_compl_of_subset
@[gcongr] theorem compl_subset_compl_of_subset (h : t ⊆ s) : sᶜ ⊆ tᶜ 
Arguments: {α : Type u} {s t : Set α} (h : t ⊆ s)
Goal: sᶜ ⊆ tᶜ
-- Set.subset_compl_iff_disjoint_left
theorem subset_compl_iff_disjoint_left : s ⊆ tᶜ ↔ Disjoint t s 
Arguments: {α : Type u} {s t : Set α}
Goal: s ⊆ tᶜ ↔ Disjoint t s
-- Set.subset_compl_iff_disjoint_right
theorem subset_compl_iff_disjoint_right : s ⊆ tᶜ ↔ Disjoint s t 
Arguments: {α : Type u} {s t : Set α}
Goal: s ⊆ tᶜ ↔ Disjoint s t
-- Set.disjoint_compl_left_iff_subset
theorem disjoint_compl_left_iff_subset : Disjoint sᶜ t ↔ t ⊆ s 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint sᶜ t ↔ t ⊆ s
-- Set.disjoint_compl_right_iff_subset
theorem disjoint_compl_right_iff_subset : Disjoint s tᶜ ↔ s ⊆ t 
Arguments: {α : Type u} {s t : Set α}
Goal: Disjoint s tᶜ ↔ s ⊆ t
-- Set.subset_union_compl_iff_inter_subset
theorem subset_union_compl_iff_inter_subset {s t u : Set α} : s ⊆ t ∪ uᶜ ↔ s ∩ u ⊆ t 
Arguments: {α : Type u} {s t u : Set α}
Goal: s ⊆ t ∪ uᶜ ↔ s ∩ u ⊆ t
-- Set.compl_subset_iff_union
theorem compl_subset_iff_union {s t : Set α} : sᶜ ⊆ t ↔ s ∪ t = univ 
Arguments: {α : Type u} {s t : Set α}
Goal: sᶜ ⊆ t ↔ s ∪ t = Set.univ
-- Set.subset_compl_singleton_iff
@[simp]
theorem subset_compl_singleton_iff {a : α} {s : Set α} : s ⊆ {a}ᶜ ↔ a ∉ s 
Arguments: {α : Type u} {a : α} {s : Set α}
Goal: s ⊆ {a}ᶜ ↔ a ∉ s
-- Set.inter_subset
theorem inter_subset (a b c : Set α) : a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c 
Arguments: {α : Type u} (a b c : Set α)
Goal: a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c
-- Set.inter_compl_nonempty_iff
theorem inter_compl_nonempty_iff {s t : Set α} : (s ∩ tᶜ).Nonempty ↔ ¬s ⊆ t 
Arguments: {α : Type u} {s t : Set α}
Goal: (s ∩ tᶜ).Nonempty ↔ ¬s ⊆ t
-- Set.not_mem_diff_of_mem
theorem not_mem_diff_of_mem {s t : Set α} {x : α} (hx : x ∈ t) : x ∉ s \ t 
Arguments: {α : Type u} {s t : Set α} {x : α} (hx : x ∈ t)
Goal: x ∉ s \ t
-- Set.mem_of_mem_diff
theorem mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∈ s 
Arguments: {α : Type u} {s t : Set α} {x : α} (h : x ∈ s \ t)
Goal: x ∈ s
-- Set.not_mem_of_mem_diff
theorem not_mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∉ t 
Arguments: {α : Type u} {s t : Set α} {x : α} (h : x ∈ s \ t)
Goal: x ∉ t
-- Set.diff_eq_compl_inter
theorem diff_eq_compl_inter {s t : Set α} : s \ t = tᶜ ∩ s 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ t = tᶜ ∩ s
-- Set.nonempty_diff
theorem nonempty_diff {s t : Set α} : (s \ t).Nonempty ↔ ¬s ⊆ t 
Arguments: {α : Type u} {s t : Set α}
Goal: (s \ t).Nonempty ↔ ¬s ⊆ t
-- Set.diff_subset
theorem diff_subset {s t : Set α} : s \ t ⊆ s 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ t ⊆ s
-- Set.diff_subset_compl
theorem diff_subset_compl (s t : Set α) : s \ t ⊆ tᶜ 
Arguments: {α : Type u} (s t : Set α)
Goal: s \ t ⊆ tᶜ
-- Set.union_diff_cancel'
theorem union_diff_cancel' {s t u : Set α} (h₁ : s ⊆ t) (h₂ : t ⊆ u) : t ∪ u \ s = u 
Arguments: {α : Type u} {s t u : Set α} (h₁ : s ⊆ t) (h₂ : t ⊆ u)
Goal: t ∪ u \ s = u
-- Set.union_diff_cancel
theorem union_diff_cancel {s t : Set α} (h : s ⊆ t) : s ∪ t \ s = t 
Arguments: {α : Type u} {s t : Set α} (h : s ⊆ t)
Goal: s ∪ t \ s = t
-- Set.union_diff_cancel_left
theorem union_diff_cancel_left {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ s = t 
Arguments: {α : Type u} {s t : Set α} (h : s ∩ t ⊆ ∅)
Goal: (s ∪ t) \ s = t
-- Set.union_diff_cancel_right
theorem union_diff_cancel_right {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ t = s 
Arguments: {α : Type u} {s t : Set α} (h : s ∩ t ⊆ ∅)
Goal: (s ∪ t) \ t = s
-- Set.union_diff_left
@[simp]
theorem union_diff_left {s t : Set α} : (s ∪ t) \ s = t \ s 
Arguments: {α : Type u} {s t : Set α}
Goal: (s ∪ t) \ s = t \ s
-- Set.union_diff_right
@[simp]
theorem union_diff_right {s t : Set α} : (s ∪ t) \ t = s \ t 
Arguments: {α : Type u} {s t : Set α}
Goal: (s ∪ t) \ t = s \ t
-- Set.union_diff_distrib
theorem union_diff_distrib {s t u : Set α} : (s ∪ t) \ u = s \ u ∪ t \ u 
Arguments: {α : Type u} {s t u : Set α}
Goal: (s ∪ t) \ u = s \ u ∪ t \ u
-- Set.inter_diff_assoc
theorem inter_diff_assoc (a b c : Set α) : (a ∩ b) \ c = a ∩ (b \ c) 
Arguments: {α : Type u} (a b c : Set α)
Goal: (a ∩ b) \ c = a ∩ (b \ c)
-- Set.inter_diff_self
@[simp]
theorem inter_diff_self (a b : Set α) : a ∩ (b \ a) = ∅ 
Arguments: {α : Type u} (a b : Set α)
Goal: a ∩ (b \ a) = ∅
-- Set.inter_union_diff
@[simp]
theorem inter_union_diff (s t : Set α) : s ∩ t ∪ s \ t = s 
Arguments: {α : Type u} (s t : Set α)
Goal: s ∩ t ∪ s \ t = s
-- Set.diff_union_inter
@[simp]
theorem diff_union_inter (s t : Set α) : s \ t ∪ s ∩ t = s 
Arguments: {α : Type u} (s t : Set α)
Goal: s \ t ∪ s ∩ t = s
-- Set.inter_union_compl
@[simp]
theorem inter_union_compl (s t : Set α) : s ∩ t ∪ s ∩ tᶜ = s 
Arguments: {α : Type u} (s t : Set α)
Goal: s ∩ t ∪ s ∩ tᶜ = s
-- Set.diff_subset_diff
@[gcongr]
theorem diff_subset_diff {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ s₂ → t₂ ⊆ t₁ → s₁ \ t₁ ⊆ s₂ \ t₂ 
Arguments: {α : Type u} {s₁ s₂ t₁ t₂ : Set α}
Goal: s₁ ⊆ s₂ → t₂ ⊆ t₁ → s₁ \ t₁ ⊆ s₂ \ t₂
-- Set.diff_subset_diff_left
@[gcongr]
theorem diff_subset_diff_left {s₁ s₂ t : Set α} (h : s₁ ⊆ s₂) : s₁ \ t ⊆ s₂ \ t 
Arguments: {α : Type u} {s₁ s₂ t : Set α} (h : s₁ ⊆ s₂)
Goal: s₁ \ t ⊆ s₂ \ t
-- Set.diff_subset_diff_right
@[gcongr]
theorem diff_subset_diff_right {s t u : Set α} (h : t ⊆ u) : s \ u ⊆ s \ t 
Arguments: {α : Type u} {s t u : Set α} (h : t ⊆ u)
Goal: s \ u ⊆ s \ t
-- Set.compl_eq_univ_diff
theorem compl_eq_univ_diff (s : Set α) : sᶜ = univ \ s 
Arguments: {α : Type u} (s : Set α)
Goal: sᶜ = Set.univ \ s
-- Set.empty_diff
@[simp]
theorem empty_diff (s : Set α) : (∅ \ s : Set α) = ∅ 
Arguments: {α : Type u} (s : Set α)
Goal: ∅ \ s = ∅
-- Set.diff_eq_empty
theorem diff_eq_empty {s t : Set α} : s \ t = ∅ ↔ s ⊆ t 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ t = ∅ ↔ s ⊆ t
-- Set.diff_empty
@[simp]
theorem diff_empty {s : Set α} : s \ ∅ = s 
Arguments: {α : Type u} {s : Set α}
Goal: s \ ∅ = s
-- Set.diff_univ
@[simp]
theorem diff_univ (s : Set α) : s \ univ = ∅ 
Arguments: {α : Type u} (s : Set α)
Goal: s \ Set.univ = ∅
-- Set.diff_diff
theorem diff_diff {u : Set α} : (s \ t) \ u = s \ (t ∪ u) 
Arguments: {α : Type u} {s t u : Set α}
Goal: (s \ t) \ u = s \ (t ∪ u)
-- Set.diff_diff_comm
theorem diff_diff_comm {s t u : Set α} : (s \ t) \ u = (s \ u) \ t 
Arguments: {α : Type u} {s t u : Set α}
Goal: (s \ t) \ u = (s \ u) \ t
-- Set.diff_subset_iff
theorem diff_subset_iff {s t u : Set α} : s \ t ⊆ u ↔ s ⊆ t ∪ u 
Arguments: {α : Type u} {s t u : Set α}
Goal: s \ t ⊆ u ↔ s ⊆ t ∪ u
-- Set.subset_diff_union
theorem subset_diff_union (s t : Set α) : s ⊆ s \ t ∪ t 
Arguments: {α : Type u} (s t : Set α)
Goal: s ⊆ s \ t ∪ t
-- Set.diff_union_of_subset
theorem diff_union_of_subset {s t : Set α} (h : t ⊆ s) : s \ t ∪ t = s 
Arguments: {α : Type u} {s t : Set α} (h : t ⊆ s)
Goal: s \ t ∪ t = s
-- Set.diff_singleton_subset_iff
@[simp]
theorem diff_singleton_subset_iff {x : α} {s t : Set α} : s \ {x} ⊆ t ↔ s ⊆ insert x t 
Arguments: {α : Type u} {x : α} {s t : Set α}
Goal: s \ {x} ⊆ t ↔ s ⊆ insert x t
-- Set.subset_diff_singleton
theorem subset_diff_singleton {x : α} {s t : Set α} (h : s ⊆ t) (hx : x ∉ s) : s ⊆ t \ {x} 
Arguments: {α : Type u} {x : α} {s t : Set α} (h : s ⊆ t) (hx : x ∉ s)
Goal: s ⊆ t \ {x}
-- Set.subset_insert_diff_singleton
theorem subset_insert_diff_singleton (x : α) (s : Set α) : s ⊆ insert x (s \ {x}) 
Arguments: {α : Type u} (x : α) (s : Set α)
Goal: s ⊆ insert x (s \ {x})
-- Set.diff_subset_comm
theorem diff_subset_comm {s t u : Set α} : s \ t ⊆ u ↔ s \ u ⊆ t 
Arguments: {α : Type u} {s t u : Set α}
Goal: s \ t ⊆ u ↔ s \ u ⊆ t
-- Set.diff_inter
theorem diff_inter {s t u : Set α} : s \ (t ∩ u) = s \ t ∪ s \ u 
Arguments: {α : Type u} {s t u : Set α}
Goal: s \ (t ∩ u) = s \ t ∪ s \ u
-- Set.diff_inter_diff
theorem diff_inter_diff {s t u : Set α} : s \ t ∩ (s \ u) = s \ (t ∪ u) 
Arguments: {α : Type u} {s t u : Set α}
Goal: s \ t ∩ (s \ u) = s \ (t ∪ u)
-- Set.diff_compl
theorem diff_compl : s \ tᶜ = s ∩ t 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ tᶜ = s ∩ t
-- Set.diff_diff_right
theorem diff_diff_right {s t u : Set α} : s \ (t \ u) = s \ t ∪ s ∩ u 
Arguments: {α : Type u} {s t u : Set α}
Goal: s \ (t \ u) = s \ t ∪ s ∩ u
-- Set.insert_diff_of_mem
@[simp]
theorem insert_diff_of_mem (s) (h : a ∈ t) : insert a s \ t = s \ t 
Arguments: {α : Type u} {a : α} {t : Set α} (s : Set α) (h : a ∈ t)
Goal: insert a s \ t = s \ t
-- Set.insert_diff_of_not_mem
theorem insert_diff_of_not_mem (s) (h : a ∉ t) : insert a s \ t = insert a (s \ t) 
Arguments: {α : Type u} {a : α} {t : Set α} (s : Set α) (h : a ∉ t)
Goal: insert a s \ t = insert a (s \ t)
-- Set.insert_diff_self_of_not_mem
theorem insert_diff_self_of_not_mem {a : α} {s : Set α} (h : a ∉ s) : insert a s \ {a} = s 
Arguments: {α : Type u} {a : α} {s : Set α} (h : a ∉ s)
Goal: insert a s \ {a} = s
-- Set.insert_diff_eq_singleton
@[simp]
theorem insert_diff_eq_singleton {a : α} {s : Set α} (h : a ∉ s) : insert a s \ s = {a} 
Arguments: {α : Type u} {a : α} {s : Set α} (h : a ∉ s)
Goal: insert a s \ s = {a}
-- Set.inter_insert_of_mem
theorem inter_insert_of_mem (h : a ∈ s) : s ∩ insert a t = insert a (s ∩ t) 
Arguments: {α : Type u} {a : α} {s t : Set α} (h : a ∈ s)
Goal: s ∩ insert a t = insert a (s ∩ t)
-- Set.insert_inter_of_mem
theorem insert_inter_of_mem (h : a ∈ t) : insert a s ∩ t = insert a (s ∩ t) 
Arguments: {α : Type u} {a : α} {s t : Set α} (h : a ∈ t)
Goal: insert a s ∩ t = insert a (s ∩ t)
-- Set.inter_insert_of_not_mem
theorem inter_insert_of_not_mem (h : a ∉ s) : s ∩ insert a t = s ∩ t 
Arguments: {α : Type u} {a : α} {s t : Set α} (h : a ∉ s)
Goal: s ∩ insert a t = s ∩ t
-- Set.insert_inter_of_not_mem
theorem insert_inter_of_not_mem (h : a ∉ t) : insert a s ∩ t = s ∩ t 
Arguments: {α : Type u} {a : α} {s t : Set α} (h : a ∉ t)
Goal: insert a s ∩ t = s ∩ t
-- Set.union_diff_self
@[simp]
theorem union_diff_self {s t : Set α} : s ∪ t \ s = s ∪ t 
Arguments: {α : Type u} {s t : Set α}
Goal: s ∪ t \ s = s ∪ t
-- Set.diff_union_self
@[simp]
theorem diff_union_self {s t : Set α} : s \ t ∪ t = s ∪ t 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ t ∪ t = s ∪ t
-- Set.diff_inter_self
@[simp]
theorem diff_inter_self {a b : Set α} : b \ a ∩ a = ∅ 
Arguments: {α : Type u} {a b : Set α}
Goal: b \ a ∩ a = ∅
-- Set.diff_inter_self_eq_diff
@[simp]
theorem diff_inter_self_eq_diff {s t : Set α} : s \ (t ∩ s) = s \ t 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ (t ∩ s) = s \ t
-- Set.diff_self_inter
@[simp]
theorem diff_self_inter {s t : Set α} : s \ (s ∩ t) = s \ t 
Arguments: {α : Type u} {s t : Set α}
Goal: s \ (s ∩ t) = s \ t
-- Set.diff_singleton_eq_self
@[simp]
theorem diff_singleton_eq_self {a : α} {s : Set α} (h : a ∉ s) : s \ {a} = s 
Arguments: {α : Type u} {a : α} {s : Set α} (h : a ∉ s)
Goal: s \ {a} = s
-- Set.diff_singleton_sSubset
@[simp]
theorem diff_singleton_sSubset {s : Set α} {a : α} : s \ {a} ⊂ s ↔ a ∈ s 
Arguments: {α : Type u} {s : Set α} {a : α}
Goal: s \ {a} ⊂ s ↔ a ∈ s
-- Set.insert_diff_singleton
@[simp]
theorem insert_diff_singleton {a : α} {s : Set α} : insert a (s \ {a}) = insert a s 
Arguments: {α : Type u} {a : α} {s : Set α}
Goal: insert a (s \ {a}) = insert a s
-- Set.insert_diff_singleton_comm
theorem insert_diff_singleton_comm (hab : a ≠ b) (s : Set α) :
    insert a (s \ {b}) = insert a s \ {b} 
Arguments: {α : Type u} {a b : α} (hab : a ≠ b) (s : Set α)
Goal: insert a (s \ {b}) = insert a s \ {b}
-- Set.diff_self
theorem diff_self {s : Set α} : s \ s = ∅ 
Arguments: {α : Type u} {s : Set α}
Goal: s \ s = ∅
-- Set.diff_diff_right_self
theorem diff_diff_right_self (s t : Set α) : s \ (s \ t) = s ∩ t 
Arguments: {α : Type u} (s t : Set α)
Goal: s \ (s \ t) = s ∩ t
-- Set.diff_diff_cancel_left
theorem diff_diff_cancel_left {s t : Set α} (h : s ⊆ t) : t \ (t \ s) = s 
Arguments: {α : Type u} {s t : Set α} (h : s ⊆ t)
Goal: t \ (t \ s) = s
-- Set.mem_diff_singleton
theorem mem_diff_singleton {x y : α} {s : Set α} : x ∈ s \ {y} ↔ x ∈ s ∧ x ≠ y 
Arguments: {α : Type u} {x y : α} {s : Set α}
Goal: x ∈ s \ {y} ↔ x ∈ s ∧ x ≠ y
-- Set.mem_diff_singleton_empty
theorem mem_diff_singleton_empty {t : Set (Set α)} : s ∈ t \ {∅} ↔ s ∈ t ∧ s.Nonempty 
Arguments: {α : Type u} {s : Set α} {t : Set (Set α)}
Goal: s ∈ t \ {∅} ↔ s ∈ t ∧ s.Nonempty
-- Set.subset_insert_iff
theorem subset_insert_iff {s t : Set α} {x : α} :
    s ⊆ insert x t ↔ s ⊆ t ∨ (x ∈ s ∧ s \ {x} ⊆ t) 
Arguments: {α : Type u} {s t : Set α} {x : α}
Goal: s ⊆ insert x t ↔ s ⊆ t ∨ x ∈ s ∧ s \ {x} ⊆ t
-- Set.union_eq_diff_union_diff_union_inter
theorem union_eq_diff_union_diff_union_inter (s t : Set α) : s ∪ t = s \ t ∪ t \ s ∪ s ∩ t 
Arguments: {α : Type u} (s t : Set α)
Goal: s ∪ t = s \ t ∪ t \ s ∪ s ∩ t
-- Set.pair_eq_singleton
theorem pair_eq_singleton (a : α) : ({a, a} : Set α) = {a} 
Arguments: {α : Type u} (a : α)
Goal: {a, a} = {a}
-- Set.pair_comm
theorem pair_comm (a b : α) : ({a, b} : Set α) = {b, a} 
Arguments: {α : Type u} (a b : α)
Goal: {a, b} = {b, a}
-- Set.pair_eq_pair_iff
theorem pair_eq_pair_iff {x y z w : α} :
    ({x, y} : Set α) = {z, w} ↔ x = z ∧ y = w ∨ x = w ∧ y = z 
Arguments: {α : Type u} {x y z w : α}
Goal: {x, y} = {z, w} ↔ x = z ∧ y = w ∨ x = w ∧ y = z
-- Set.pair_diff_left
theorem pair_diff_left (hne : a ≠ b) : ({a, b} : Set α) \ {a} = {b} 
Arguments: {α : Type u} {a b : α} (hne : a ≠ b)
Goal: {a, b} \ {a} = {b}
-- Set.pair_diff_right
theorem pair_diff_right (hne : a ≠ b) : ({a, b} : Set α) \ {b} = {a} 
Arguments: {α : Type u} {a b : α} (hne : a ≠ b)
Goal: {a, b} \ {b} = {a}
-- Set.pair_subset_iff
theorem pair_subset_iff : {a, b} ⊆ s ↔ a ∈ s ∧ b ∈ s 
Arguments: {α : Type u} {a b : α} {s : Set α}
Goal: {a, b} ⊆ s ↔ a ∈ s ∧ b ∈ s
-- Set.pair_subset
theorem pair_subset (ha : a ∈ s) (hb : b ∈ s) : {a, b} ⊆ s 
Arguments: {α : Type u} {a b : α} {s : Set α} (ha : a ∈ s) (hb : b ∈ s)
Goal: {a, b} ⊆ s
-- Set.subset_pair_iff
theorem subset_pair_iff : s ⊆ {a, b} ↔ ∀ x ∈ s, x = a ∨ x = b 
Arguments: {α : Type u} {a b : α} {s : Set α}
Goal: s ⊆ {a, b} ↔ ∀ x ∈ s, x = a ∨ x = b
-- Set.subset_pair_iff_eq
theorem subset_pair_iff_eq {x y : α} : s ⊆ {x, y} ↔ s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y} 
Arguments: {α : Type u} {s : Set α} {x y : α}
Goal: s ⊆ {x, y} ↔ s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y}
-- Set.Nonempty.subset_pair_iff_eq
theorem Nonempty.subset_pair_iff_eq (hs : s.Nonempty) :
    s ⊆ {a, b} ↔ s = {a} ∨ s = {b} ∨ s = {a, b} 
Arguments: {α : Type u} {a b : α} {s : Set α} (hs : s.Nonempty)
Goal: s ⊆ {a, b} ↔ s = {a} ∨ s = {b} ∨ s = {a, b}
-- Set.mem_symmDiff
theorem mem_symmDiff : a ∈ s ∆ t ↔ a ∈ s ∧ a ∉ t ∨ a ∈ t ∧ a ∉ s 
Arguments: {α : Type u} {a : α} {s t : Set α}
Goal: a ∈ symmDiff s t ↔ a ∈ s ∧ a ∉ t ∨ a ∈ t ∧ a ∉ s
-- Set.symmDiff_def
protected theorem symmDiff_def (s t : Set α) : s ∆ t = s \ t ∪ t \ s 
Arguments: {α : Type u} (s t : Set α)
Goal: symmDiff s t = s \ t ∪ t \ s