# Infinitude of intervals #

Bounded intervals in dense orders are infinite, as are unbounded intervals in orders that are unbounded on the appropriate side. We also prove that an unbounded preorder is an infinite type.

source
instance NoMaxOrder.infinite {α : Type u_1} [Preorder α] [Nonempty α] [NoMaxOrder α] :
Infinite α

A nonempty preorder with no maximal element is infinite.

source
instance NoMinOrder.infinite {α : Type u_1} [Preorder α] [Nonempty α] [NoMinOrder α] :
Infinite α

A nonempty preorder with no minimal element is infinite.

source
theorem Set.Ioo.infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
Infinite ↑(Ioo a b)
source
theorem Set.Ioo_infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
(Ioo a b).Infinite
source
theorem Set.Ico_infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
(Ico a b).Infinite
source
theorem Set.Ico.infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
Infinite ↑(Ico a b)
source
theorem Set.Ioc_infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
(Ioc a b).Infinite
source
theorem Set.Ioc.infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
Infinite ↑(Ioc a b)
source
theorem Set.Icc_infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
(Icc a b).Infinite
source
theorem Set.Icc.infinite {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) :
Infinite ↑(Icc a b)
source
instance Set.instInfiniteElemIioOfNoMinOrder {α : Type u_1} [Preorder α] [NoMinOrder α] {a : α} :
Infinite ↑(Iio a)
source
theorem Set.Iio_infinite {α : Type u_1} [Preorder α] [NoMinOrder α] (a : α) :
(Iio a).Infinite
source
instance Set.instInfiniteElemIicOfNoMinOrder {α : Type u_1} [Preorder α] [NoMinOrder α] {a : α} :
Infinite ↑(Iic a)
source
theorem Set.Iic_infinite {α : Type u_1} [Preorder α] [NoMinOrder α] (a : α) :
(Iic a).Infinite
source
instance Set.instInfiniteElemIoiOfNoMaxOrder {α : Type u_1} [Preorder α] [NoMaxOrder α] {a : α} :
Infinite ↑(Ioi a)
source
theorem Set.Ioi_infinite {α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α) :
(Ioi a).Infinite
source
instance Set.instInfiniteElemIciOfNoMaxOrder {α : Type u_1} [Preorder α] [NoMaxOrder α] {a : α} :
Infinite ↑(Ici a)
source
theorem Set.Ici_infinite {α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α) :
(Ici a).Infinite