File: Mathlib/Order/Bounded.lean

full_name: Set.Bounded.mono
code: theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounded r s 
goal: Set.Bounded r s

full_name: Set.Unbounded.mono
code: theorem Unbounded.mono (hst : s ⊆ t) (hs : Unbounded r s) : Unbounded r t 
goal: Set.Unbounded r t

full_name: Set.unbounded_le_of_forall_exists_lt
code: theorem unbounded_le_of_forall_exists_lt [Preorder α] (h : ∀ a, ∃ b ∈ s, a < b) :
    Unbounded (· ≤ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) s

full_name: Set.unbounded_le_iff
code: theorem unbounded_le_iff [LinearOrder α] : Unbounded (· ≤ ·) s ↔ ∀ a, ∃ b ∈ s, a < b 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) s ↔ ∀ (a : α), ∃ b ∈ s, a < b

full_name: Set.unbounded_lt_of_forall_exists_le
code: theorem unbounded_lt_of_forall_exists_le [Preorder α] (h : ∀ a, ∃ b ∈ s, a ≤ b) :
    Unbounded (· < ·) s 
goal: Set.Unbounded (fun x x_1 => x < x_1) s

full_name: Set.unbounded_lt_iff
code: theorem unbounded_lt_iff [LinearOrder α] : Unbounded (· < ·) s ↔ ∀ a, ∃ b ∈ s, a ≤ b 
goal: Set.Unbounded (fun x x_1 => x < x_1) s ↔ ∀ (a : α), ∃ b ∈ s, a ≤ b

full_name: Set.unbounded_ge_of_forall_exists_gt
code: theorem unbounded_ge_of_forall_exists_gt [Preorder α] (h : ∀ a, ∃ b ∈ s, b < a) :
    Unbounded (· ≥ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≥ x_1) s

full_name: Set.unbounded_ge_iff
code: theorem unbounded_ge_iff [LinearOrder α] : Unbounded (· ≥ ·) s ↔ ∀ a, ∃ b ∈ s, b < a 
goal: Set.Unbounded (fun x x_1 => x ≥ x_1) s ↔ ∀ (a : α), ∃ b ∈ s, b < a

full_name: Set.unbounded_gt_of_forall_exists_ge
code: theorem unbounded_gt_of_forall_exists_ge [Preorder α] (h : ∀ a, ∃ b ∈ s, b ≤ a) :
    Unbounded (· > ·) s 
goal: Set.Unbounded (fun x x_1 => x > x_1) s

full_name: Set.unbounded_gt_iff
code: theorem unbounded_gt_iff [LinearOrder α] : Unbounded (· > ·) s ↔ ∀ a, ∃ b ∈ s, b ≤ a 
goal: Set.Unbounded (fun x x_1 => x > x_1) s ↔ ∀ (a : α), ∃ b ∈ s, b ≤ a

full_name: Set.Bounded.rel_mono
code: theorem Bounded.rel_mono {r' : α → α → Prop} (h : Bounded r s) (hrr' : r ≤ r') : Bounded r' s 
goal: Set.Bounded r' s

full_name: Set.bounded_le_of_bounded_lt
code: theorem bounded_le_of_bounded_lt [Preorder α] (h : Bounded (· < ·) s) : Bounded (· ≤ ·) s 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) s

full_name: Set.Unbounded.rel_mono
code: theorem Unbounded.rel_mono {r' : α → α → Prop} (hr : r' ≤ r) (h : Unbounded r s) : Unbounded r' s 
goal: Set.Unbounded r' s

full_name: Set.unbounded_lt_of_unbounded_le
code: theorem unbounded_lt_of_unbounded_le [Preorder α] (h : Unbounded (· ≤ ·) s) : Unbounded (· < ·) s 
goal: Set.Unbounded (fun x x_1 => x < x_1) s

full_name: Set.bounded_le_iff_bounded_lt
code: theorem bounded_le_iff_bounded_lt [Preorder α] [NoMaxOrder α] :
    Bounded (· ≤ ·) s ↔ Bounded (· < ·) s 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) s ↔ Set.Bounded (fun x x_1 => x < x_1) s

full_name: Set.unbounded_lt_iff_unbounded_le
code: theorem unbounded_lt_iff_unbounded_le [Preorder α] [NoMaxOrder α] :
    Unbounded (· < ·) s ↔ Unbounded (· ≤ ·) s 
goal: Set.Unbounded (fun x x_1 => x < x_1) s ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s

full_name: Set.bounded_ge_of_bounded_gt
code: theorem bounded_ge_of_bounded_gt [Preorder α] (h : Bounded (· > ·) s) : Bounded (· ≥ ·) s 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) s

full_name: Set.unbounded_gt_of_unbounded_ge
code: theorem unbounded_gt_of_unbounded_ge [Preorder α] (h : Unbounded (· ≥ ·) s) : Unbounded (· > ·) s 
goal: Set.Unbounded (fun x x_1 => x > x_1) s

full_name: Set.bounded_ge_iff_bounded_gt
code: theorem bounded_ge_iff_bounded_gt [Preorder α] [NoMinOrder α] :
    Bounded (· ≥ ·) s ↔ Bounded (· > ·) s 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) s ↔ Set.Bounded (fun x x_1 => x > x_1) s

full_name: Set.unbounded_gt_iff_unbounded_ge
code: theorem unbounded_gt_iff_unbounded_ge [Preorder α] [NoMinOrder α] :
    Unbounded (· > ·) s ↔ Unbounded (· ≥ ·) s 
goal: Set.Unbounded (fun x x_1 => x > x_1) s ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s

full_name: Set.unbounded_le_univ
code: theorem unbounded_le_univ [LE α] [NoTopOrder α] : Unbounded (· ≤ ·) (@Set.univ α) 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) Set.univ

full_name: Set.unbounded_lt_univ
code: theorem unbounded_lt_univ [Preorder α] [NoTopOrder α] : Unbounded (· < ·) (@Set.univ α) 
goal: Set.Unbounded (fun x x_1 => x < x_1) Set.univ

full_name: Set.unbounded_ge_univ
code: theorem unbounded_ge_univ [LE α] [NoBotOrder α] : Unbounded (· ≥ ·) (@Set.univ α) 
goal: Set.Unbounded (fun x x_1 => x ≥ x_1) Set.univ

full_name: Set.unbounded_gt_univ
code: theorem unbounded_gt_univ [Preorder α] [NoBotOrder α] : Unbounded (· > ·) (@Set.univ α) 
goal: Set.Unbounded (fun x x_1 => x > x_1) Set.univ

full_name: Set.bounded_self
code: theorem bounded_self (a : α) : Bounded r { b | r b a } 
goal: Set.Bounded r {b | r b a}

full_name: Set.bounded_lt_Iio
code: theorem bounded_lt_Iio [Preorder α] (a : α) : Bounded (· < ·) (Iio a) 
goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Iio a)

full_name: Set.bounded_le_Iio
code: theorem bounded_le_Iio [Preorder α] (a : α) : Bounded (· ≤ ·) (Iio a) 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Iio a)

full_name: Set.bounded_le_Iic
code: theorem bounded_le_Iic [Preorder α] (a : α) : Bounded (· ≤ ·) (Iic a) 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Iic a)

full_name: Set.bounded_lt_Iic
code: theorem bounded_lt_Iic [Preorder α] [NoMaxOrder α] (a : α) : Bounded (· < ·) (Iic a) 
goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Iic a)

full_name: Set.bounded_gt_Ioi
code: theorem bounded_gt_Ioi [Preorder α] (a : α) : Bounded (· > ·) (Ioi a) 
goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ioi a)

full_name: Set.bounded_ge_Ioi
code: theorem bounded_ge_Ioi [Preorder α] (a : α) : Bounded (· ≥ ·) (Ioi a) 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ioi a)

full_name: Set.bounded_ge_Ici
code: theorem bounded_ge_Ici [Preorder α] (a : α) : Bounded (· ≥ ·) (Ici a) 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ici a)

full_name: Set.bounded_gt_Ici
code: theorem bounded_gt_Ici [Preorder α] [NoMinOrder α] (a : α) : Bounded (· > ·) (Ici a) 
goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ici a)

full_name: Set.bounded_lt_Ioo
code: theorem bounded_lt_Ioo [Preorder α] (a b : α) : Bounded (· < ·) (Ioo a b) 
goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Ioo a b)

full_name: Set.bounded_lt_Ico
code: theorem bounded_lt_Ico [Preorder α] (a b : α) : Bounded (· < ·) (Ico a b) 
goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Ico a b)

full_name: Set.bounded_lt_Ioc
code: theorem bounded_lt_Ioc [Preorder α] [NoMaxOrder α] (a b : α) : Bounded (· < ·) (Ioc a b) 
goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Ioc a b)

full_name: Set.bounded_lt_Icc
code: theorem bounded_lt_Icc [Preorder α] [NoMaxOrder α] (a b : α) : Bounded (· < ·) (Icc a b) 
goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Icc a b)

full_name: Set.bounded_le_Ioo
code: theorem bounded_le_Ioo [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ioo a b) 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Ioo a b)

full_name: Set.bounded_le_Ico
code: theorem bounded_le_Ico [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ico a b) 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Ico a b)

full_name: Set.bounded_le_Ioc
code: theorem bounded_le_Ioc [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ioc a b) 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Ioc a b)

full_name: Set.bounded_le_Icc
code: theorem bounded_le_Icc [Preorder α] (a b : α) : Bounded (· ≤ ·) (Icc a b) 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Icc a b)

full_name: Set.bounded_gt_Ioo
code: theorem bounded_gt_Ioo [Preorder α] (a b : α) : Bounded (· > ·) (Ioo a b) 
goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ioo a b)

full_name: Set.bounded_gt_Ioc
code: theorem bounded_gt_Ioc [Preorder α] (a b : α) : Bounded (· > ·) (Ioc a b) 
goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ioc a b)

full_name: Set.bounded_gt_Ico
code: theorem bounded_gt_Ico [Preorder α] [NoMinOrder α] (a b : α) : Bounded (· > ·) (Ico a b) 
goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ico a b)

full_name: Set.bounded_gt_Icc
code: theorem bounded_gt_Icc [Preorder α] [NoMinOrder α] (a b : α) : Bounded (· > ·) (Icc a b) 
goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Icc a b)

full_name: Set.bounded_ge_Ioo
code: theorem bounded_ge_Ioo [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ioo a b) 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ioo a b)

full_name: Set.bounded_ge_Ioc
code: theorem bounded_ge_Ioc [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ioc a b) 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ioc a b)

full_name: Set.bounded_ge_Ico
code: theorem bounded_ge_Ico [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ico a b) 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ico a b)

full_name: Set.bounded_ge_Icc
code: theorem bounded_ge_Icc [Preorder α] (a b : α) : Bounded (· ≥ ·) (Icc a b) 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Icc a b)

full_name: Set.unbounded_le_Ioi
code: theorem unbounded_le_Ioi [SemilatticeSup α] [NoMaxOrder α] (a : α) :
    Unbounded (· ≤ ·) (Ioi a) 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (Set.Ioi a)

full_name: Set.unbounded_le_Ici
code: theorem unbounded_le_Ici [SemilatticeSup α] [NoMaxOrder α] (a : α) :
    Unbounded (· ≤ ·) (Ici a) 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (Set.Ici a)

full_name: Set.unbounded_lt_Ioi
code: theorem unbounded_lt_Ioi [SemilatticeSup α] [NoMaxOrder α] (a : α) :
    Unbounded (· < ·) (Ioi a) 
goal: Set.Unbounded (fun x x_1 => x < x_1) (Set.Ioi a)

full_name: Set.unbounded_lt_Ici
code: theorem unbounded_lt_Ici [SemilatticeSup α] (a : α) : Unbounded (· < ·) (Ici a) 
goal: Set.Unbounded (fun x x_1 => x < x_1) (Set.Ici a)

full_name: Set.bounded_inter_not
code: theorem bounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
    Bounded r (s ∩ { b | ¬r b a }) ↔ Bounded r s 
goal: Set.Bounded r (s ∩ {b | ¬r b a}) ↔ Set.Bounded r s

full_name: Set.unbounded_inter_not
code: theorem unbounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
    Unbounded r (s ∩ { b | ¬r b a }) ↔ Unbounded r s 
goal: Set.Unbounded r (s ∩ {b | ¬r b a}) ↔ Set.Unbounded r s

full_name: Set.bounded_le_inter_not_le
code: theorem bounded_le_inter_not_le [SemilatticeSup α] (a : α) :
    Bounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Bounded (· ≤ ·) s 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Set.Bounded (fun x x_1 => x ≤ x_1) s

full_name: Set.unbounded_le_inter_not_le
code: theorem unbounded_le_inter_not_le [SemilatticeSup α] (a : α) :
    Unbounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Unbounded (· ≤ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s

full_name: Set.bounded_le_inter_lt
code: theorem bounded_le_inter_lt [LinearOrder α] (a : α) :
    Bounded (· ≤ ·) (s ∩ { b | a < b }) ↔ Bounded (· ≤ ·) s 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b}) ↔ Set.Bounded (fun x x_1 => x ≤ x_1) s

full_name: Set.unbounded_le_inter_lt
code: theorem unbounded_le_inter_lt [LinearOrder α] (a : α) :
    Unbounded (· ≤ ·) (s ∩ { b | a < b }) ↔ Unbounded (· ≤ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s

full_name: Set.bounded_le_inter_le
code: theorem bounded_le_inter_le [LinearOrder α] (a : α) :
    Bounded (· ≤ ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· ≤ ·) s 
goal: Set.Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Set.Bounded (fun x x_1 => x ≤ x_1) s

full_name: Set.unbounded_le_inter_le
code: theorem unbounded_le_inter_le [LinearOrder α] (a : α) :
    Unbounded (· ≤ ·) (s ∩ { b | a ≤ b }) ↔ Unbounded (· ≤ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s

full_name: Set.bounded_lt_inter_not_lt
code: theorem bounded_lt_inter_not_lt [SemilatticeSup α] (a : α) :
    Bounded (· < ·) (s ∩ { b | ¬b < a }) ↔ Bounded (· < ·) s 
goal: Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | ¬b < a}) ↔ Set.Bounded (fun x x_1 => x < x_1) s

full_name: Set.unbounded_lt_inter_not_lt
code: theorem unbounded_lt_inter_not_lt [SemilatticeSup α] (a : α) :
    Unbounded (· < ·) (s ∩ { b | ¬b < a }) ↔ Unbounded (· < ·) s 
goal: Set.Unbounded (fun x x_1 => x < x_1) (s ∩ {b | ¬b < a}) ↔ Set.Unbounded (fun x x_1 => x < x_1) s

full_name: Set.bounded_lt_inter_le
code: theorem bounded_lt_inter_le [LinearOrder α] (a : α) :
    Bounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· < ·) s 
goal: Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | a ≤ b}) ↔ Set.Bounded (fun x x_1 => x < x_1) s

full_name: Set.unbounded_lt_inter_le
code: theorem unbounded_lt_inter_le [LinearOrder α] (a : α) :
    Unbounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Unbounded (· < ·) s 
goal: Set.Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x < x_1) s

full_name: Set.bounded_lt_inter_lt
code: theorem bounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
    Bounded (· < ·) (s ∩ { b | a < b }) ↔ Bounded (· < ·) s 
goal: Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Set.Bounded (fun x x_1 => x < x_1) s

full_name: Set.unbounded_lt_inter_lt
code: theorem unbounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
    Unbounded (· < ·) (s ∩ { b | a < b }) ↔ Unbounded (· < ·) s 
goal: Set.Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Set.Unbounded (fun x x_1 => x < x_1) s

full_name: Set.bounded_ge_inter_not_ge
code: theorem bounded_ge_inter_not_ge [SemilatticeInf α] (a : α) :
    Bounded (· ≥ ·) (s ∩ { b | ¬a ≤ b }) ↔ Bounded (· ≥ ·) s 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (s ∩ {b | ¬a ≤ b}) ↔ Set.Bounded (fun x x_1 => x ≥ x_1) s

full_name: Set.unbounded_ge_inter_not_ge
code: theorem unbounded_ge_inter_not_ge [SemilatticeInf α] (a : α) :
    Unbounded (· ≥ ·) (s ∩ { b | ¬a ≤ b }) ↔ Unbounded (· ≥ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≥ x_1) (s ∩ {b | ¬a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s

full_name: Set.bounded_ge_inter_gt
code: theorem bounded_ge_inter_gt [LinearOrder α] (a : α) :
    Bounded (· ≥ ·) (s ∩ { b | b < a }) ↔ Bounded (· ≥ ·) s 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b < a}) ↔ Set.Bounded (fun x x_1 => x ≥ x_1) s

full_name: Set.unbounded_ge_inter_gt
code: theorem unbounded_ge_inter_gt [LinearOrder α] (a : α) :
    Unbounded (· ≥ ·) (s ∩ { b | b < a }) ↔ Unbounded (· ≥ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b < a}) ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s

full_name: Set.bounded_ge_inter_ge
code: theorem bounded_ge_inter_ge [LinearOrder α] (a : α) :
    Bounded (· ≥ ·) (s ∩ { b | b ≤ a }) ↔ Bounded (· ≥ ·) s 
goal: Set.Bounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b ≤ a}) ↔ Set.Bounded (fun x x_1 => x ≥ x_1) s

full_name: Set.unbounded_ge_iff_unbounded_inter_ge
code: theorem unbounded_ge_iff_unbounded_inter_ge [LinearOrder α] (a : α) :
    Unbounded (· ≥ ·) (s ∩ { b | b ≤ a }) ↔ Unbounded (· ≥ ·) s 
goal: Set.Unbounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b ≤ a}) ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s

full_name: Set.bounded_gt_inter_not_gt
code: theorem bounded_gt_inter_not_gt [SemilatticeInf α] (a : α) :
    Bounded (· > ·) (s ∩ { b | ¬a < b }) ↔ Bounded (· > ·) s 
goal: Set.Bounded (fun x x_1 => x > x_1) (s ∩ {b | ¬a < b}) ↔ Set.Bounded (fun x x_1 => x > x_1) s

full_name: Set.unbounded_gt_inter_not_gt
code: theorem unbounded_gt_inter_not_gt [SemilatticeInf α] (a : α) :
    Unbounded (· > ·) (s ∩ { b | ¬a < b }) ↔ Unbounded (· > ·) s 
goal: Set.Unbounded (fun x x_1 => x > x_1) (s ∩ {b | ¬a < b}) ↔ Set.Unbounded (fun x x_1 => x > x_1) s

full_name: Set.bounded_gt_inter_ge
code: theorem bounded_gt_inter_ge [LinearOrder α] (a : α) :
    Bounded (· > ·) (s ∩ { b | b ≤ a }) ↔ Bounded (· > ·) s 
goal: Set.Bounded (fun x x_1 => x > x_1) (s ∩ {b | b ≤ a}) ↔ Set.Bounded (fun x x_1 => x > x_1) s

full_name: Set.unbounded_inter_ge
code: theorem unbounded_inter_ge [LinearOrder α] (a : α) :
    Unbounded (· > ·) (s ∩ { b | b ≤ a }) ↔ Unbounded (· > ·) s 
goal: Set.Unbounded (fun x x_1 => x > x_1) (s ∩ {b | b ≤ a}) ↔ Set.Unbounded (fun x x_1 => x > x_1) s

full_name: Set.bounded_gt_inter_gt
code: theorem bounded_gt_inter_gt [LinearOrder α] [NoMinOrder α] (a : α) :
    Bounded (· > ·) (s ∩ { b | b < a }) ↔ Bounded (· > ·) s 
goal: Set.Bounded (fun x x_1 => x > x_1) (s ∩ {b | b < a}) ↔ Set.Bounded (fun x x_1 => x > x_1) s

full_name: Set.unbounded_gt_inter_gt
code: theorem unbounded_gt_inter_gt [LinearOrder α] [NoMinOrder α] (a : α) :
    Unbounded (· > ·) (s ∩ { b | b < a }) ↔ Unbounded (· > ·) s 
goal: Set.Unbounded (fun x x_1 => x > x_1) (s ∩ {b | b < a}) ↔ Set.Unbounded (fun x x_1 => x > x_1) s