File: Mathlib/Topology/Order/LeftRightNhds.lean

full_name: TFAE_mem_nhdsWithin_Ioi
code: /-- The following statements are equivalent:

0. `s` is a neighborhood of `a` within `(a, +∞)`;
1. `s` is a neighborhood of `a` within `(a, b]`;
2. `s` is a neighborhood of `a` within `(a, b)`;
3. `s` includes `(a, u)` for some `u ∈ (a, b]`;
4. `s` includes `(a, u)` for some `u > a`.
-/
theorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[>] a,
      s ∈ 𝓝[Ioc a b] a,
      s ∈ 𝓝[Ioo a b] a,
      ∃ u ∈ Ioc a b, Ioo a u ⊆ s,
      ∃ u ∈ Ioi a, Ioo a u ⊆ s] 
goal: [s ∈ 𝓝[>] a, s ∈ 𝓝[Set.Ioc a b] a, s ∈ 𝓝[Set.Ioo a b] a, ∃ u ∈ Set.Ioc a b, Set.Ioo a u ⊆ s,
    ∃ u ∈ Set.Ioi a, Set.Ioo a u ⊆ s].TFAE

full_name: mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset
code: theorem mem_nhdsWithin_Ioi_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s 
goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioc a u', Set.Ioo a u ⊆ s

full_name: mem_nhdsWithin_Ioi_iff_exists_Ioo_subset'
code: /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u < u'`, provided `a` is not a top element. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s 
goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioo a u ⊆ s

full_name: nhdsWithin_Ioi_basis'
code: theorem nhdsWithin_Ioi_basis' {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) 
goal: (𝓝[>] a).HasBasis (fun x => a < x) (Set.Ioo a)

full_name: nhdsWithin_Ioi_basis
code: lemma nhdsWithin_Ioi_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) 
goal: (𝓝[>] a).HasBasis (fun x => a < x) (Set.Ioo a)

full_name: nhdsWithin_Ioi_eq_bot_iff
code: theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b 
goal: 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b

full_name: mem_nhdsWithin_Ioi_iff_exists_Ioo_subset
code: /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s 
goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioo a u ⊆ s

full_name: countable_setOf_isolated_right
code: /-- The set of points which are isolated on the right is countable when the space is
second-countable. -/
theorem countable_setOf_isolated_right [SecondCountableTopology α] :
    { x : α | 𝓝[>] x = ⊥ }.Countable 
goal: {x | 𝓝[>] x = ⊥}.Countable

full_name: countable_setOf_isolated_left
code: /-- The set of points which are isolated on the left is countable when the space is
second-countable. -/
theorem countable_setOf_isolated_left [SecondCountableTopology α] :
    { x : α | 𝓝[<] x = ⊥ }.Countable 
goal: {x | 𝓝[<] x = ⊥}.Countable

full_name: mem_nhdsWithin_Ioi_iff_exists_Ioc_subset
code: /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ioi_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s 
goal: s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioc a u ⊆ s

full_name: TFAE_mem_nhdsWithin_Iio
code: /-- The following statements are equivalent:

0. `s` is a neighborhood of `b` within `(-∞, b)`
1. `s` is a neighborhood of `b` within `[a, b)`
2. `s` is a neighborhood of `b` within `(a, b)`
3. `s` includes `(l, b)` for some `l ∈ [a, b)`
4. `s` includes `(l, b)` for some `l < b` -/
theorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
        s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`
        s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`
        ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)`
        ∃ l ∈ Iio b, Ioo l b ⊆ s] 
goal: [s ∈ 𝓝[<] b, s ∈ 𝓝[Set.Ico a b] b, s ∈ 𝓝[Set.Ioo a b] b, ∃ l ∈ Set.Ico a b, Set.Ioo l b ⊆ s,
    ∃ l ∈ Set.Iio b, Set.Ioo l b ⊆ s].TFAE

full_name: mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset
code: theorem mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s 
goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Ico l' a, Set.Ioo l a ⊆ s

full_name: mem_nhdsWithin_Iio_iff_exists_Ioo_subset'
code: /-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`, provided `a` is not a bottom element. -/
theorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s 
goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Iio a, Set.Ioo l a ⊆ s

full_name: mem_nhdsWithin_Iio_iff_exists_Ioo_subset
code: /-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)`
with `l < a`. -/
theorem mem_nhdsWithin_Iio_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s 
goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Iio a, Set.Ioo l a ⊆ s

full_name: mem_nhdsWithin_Iio_iff_exists_Ico_subset
code: /-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)`
with `l < a`. -/
theorem mem_nhdsWithin_Iio_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s 
goal: s ∈ 𝓝[<] a ↔ ∃ l ∈ Set.Iio a, Set.Ico l a ⊆ s

full_name: nhdsWithin_Iio_basis'
code: theorem nhdsWithin_Iio_basis' {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) 
goal: (𝓝[<] a).HasBasis (fun x => x < a) fun x => Set.Ioo x a

full_name: nhdsWithin_Iio_eq_bot_iff
code: theorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a 
goal: 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a

full_name: TFAE_mem_nhdsWithin_Ici
code: /-- The following statements are equivalent:

0. `s` is a neighborhood of `a` within `[a, +∞)`;
1. `s` is a neighborhood of `a` within `[a, b]`;
2. `s` is a neighborhood of `a` within `[a, b)`;
3. `s` includes `[a, u)` for some `u ∈ (a, b]`;
4. `s` includes `[a, u)` for some `u > a`.
-/
theorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[≥] a,
      s ∈ 𝓝[Icc a b] a,
      s ∈ 𝓝[Ico a b] a,
      ∃ u ∈ Ioc a b, Ico a u ⊆ s,
      ∃ u ∈ Ioi a , Ico a u ⊆ s] 
goal: [s ∈ 𝓝[≥] a, s ∈ 𝓝[Set.Icc a b] a, s ∈ 𝓝[Set.Ico a b] a, ∃ u ∈ Set.Ioc a b, Set.Ico a u ⊆ s,
    ∃ u ∈ Set.Ioi a, Set.Ico a u ⊆ s].TFAE

full_name: mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset
code: theorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s 
goal: s ∈ 𝓝[≥] a ↔ ∃ u ∈ Set.Ioc a u', Set.Ico a u ⊆ s

full_name: mem_nhdsWithin_Ici_iff_exists_Ico_subset'
code: /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u < u'`, provided `a` is not a top element. -/
theorem mem_nhdsWithin_Ici_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') :
    s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s 
goal: s ∈ 𝓝[≥] a ↔ ∃ u ∈ Set.Ioi a, Set.Ico a u ⊆ s

full_name: mem_nhdsWithin_Ici_iff_exists_Ico_subset
code: /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)`
with `a < u`. -/
theorem mem_nhdsWithin_Ici_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s 
goal: s ∈ 𝓝[≥] a ↔ ∃ u ∈ Set.Ioi a, Set.Ico a u ⊆ s

full_name: nhdsWithin_Ici_basis_Ico
code: theorem nhdsWithin_Ici_basis_Ico [NoMaxOrder α] (a : α) :
    (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) 
goal: (𝓝[≥] a).HasBasis (fun u => a < u) (Set.Ico a)

full_name: nhdsWithin_Ici_basis_Icc
code: /-- The filter of right neighborhoods has a basis of closed intervals. -/
theorem nhdsWithin_Ici_basis_Icc [NoMaxOrder α] [DenselyOrdered α] {a : α} :
    (𝓝[≥] a).HasBasis (a < ·) (Icc a) 
goal: (𝓝[≥] a).HasBasis (fun x => a < x) (Set.Icc a)

full_name: mem_nhdsWithin_Ici_iff_exists_Icc_subset
code: /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. -/
theorem mem_nhdsWithin_Ici_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s 
goal: s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Set.Icc a u ⊆ s

full_name: TFAE_mem_nhdsWithin_Iic
code: /-- The following statements are equivalent:

0. `s` is a neighborhood of `b` within `(-∞, b]`
1. `s` is a neighborhood of `b` within `[a, b]`
2. `s` is a neighborhood of `b` within `(a, b]`
3. `s` includes `(l, b]` for some `l ∈ [a, b)`
4. `s` includes `(l, b]` for some `l < b` -/
theorem TFAE_mem_nhdsWithin_Iic {a b : α} (h : a < b) (s : Set α) :
    TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]`
      s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]`
      s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]`
      ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)`
      ∃ l ∈ Iio b, Ioc l b ⊆ s] 
goal: [s ∈ 𝓝[≤] b, s ∈ 𝓝[Set.Icc a b] b, s ∈ 𝓝[Set.Ioc a b] b, ∃ l ∈ Set.Ico a b, Set.Ioc l b ⊆ s,
    ∃ l ∈ Set.Iio b, Set.Ioc l b ⊆ s].TFAE

full_name: mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset
code: theorem mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s 
goal: s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Ico l' a, Set.Ioc l a ⊆ s

full_name: mem_nhdsWithin_Iic_iff_exists_Ioc_subset'
code: /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`, provided `a` is not a bottom element. -/
theorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) :
    s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s 
goal: s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Iio a, Set.Ioc l a ⊆ s

full_name: mem_nhdsWithin_Iic_iff_exists_Ioc_subset
code: /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]`
with `l < a`. -/
theorem mem_nhdsWithin_Iic_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} :
    s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s 
goal: s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Iio a, Set.Ioc l a ⊆ s

full_name: mem_nhdsWithin_Iic_iff_exists_Icc_subset
code: /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
with `l < a`. -/
theorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}
    {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s 
goal: s ∈ 𝓝[≤] a ↔ ∃ l < a, Set.Icc l a ⊆ s

full_name: nhdsWithin_Iic_basis_Icc
code: /-- The filter of left neighborhoods has a basis of closed intervals. -/
theorem nhdsWithin_Iic_basis_Icc [NoMinOrder α] [DenselyOrdered α] {a : α} :
    (𝓝[≤] a).HasBasis (· < a) (Icc · a) 
goal: (𝓝[≤] a).HasBasis (fun x => x < a) fun x => Set.Icc x a

full_name: nhds_eq_iInf_abs_sub
code: theorem nhds_eq_iInf_abs_sub (a : α) : 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r } 
goal: 𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}

full_name: orderTopology_of_nhds_abs
code: theorem orderTopology_of_nhds_abs {α : Type*} [TopologicalSpace α] [LinearOrderedAddCommGroup α]
    (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 0, 𝓟 { b | |a - b| < r }) : OrderTopology α 
goal: OrderTopology α

full_name: LinearOrderedAddCommGroup.tendsto_nhds
code: theorem LinearOrderedAddCommGroup.tendsto_nhds {x : Filter β} {a : α} :
    Tendsto f x (𝓝 a) ↔ ∀ ε > (0 : α), ∀ᶠ b in x, |f b - a| < ε 
goal: Filter.Tendsto f x (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ (b : β) in x, |f b - a| < ε

full_name: eventually_abs_sub_lt
code: theorem eventually_abs_sub_lt (a : α) {ε : α} (hε : 0 < ε) : ∀ᶠ x in 𝓝 a, |x - a| < ε 
goal: ∀ᶠ (x : α) in 𝓝 a, |x - a| < ε

full_name: Filter.Tendsto.add_atTop
code: /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `atTop` then `f + g` tends to `atTop`. -/
theorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x + g x) l atTop 
goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop

full_name: Filter.Tendsto.add_atBot
code: /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C`
and `g` tends to `atBot` then `f + g` tends to `atBot`. -/
theorem Filter.Tendsto.add_atBot {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) :
    Tendsto (fun x => f x + g x) l atBot 
goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot

full_name: Filter.Tendsto.atTop_add
code: /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to
`atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/
theorem Filter.Tendsto.atTop_add {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :
    Tendsto (fun x => f x + g x) l atTop 
goal: Filter.Tendsto (fun x => f x + g x) l Filter.atTop

full_name: Filter.Tendsto.atBot_add
code: /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to
`atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/
theorem Filter.Tendsto.atBot_add {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) :
    Tendsto (fun x => f x + g x) l atBot 
goal: Filter.Tendsto (fun x => f x + g x) l Filter.atBot

full_name: nhds_basis_abs_sub_lt
code: theorem nhds_basis_abs_sub_lt [NoMaxOrder α] (a : α) :
    (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b - a| < ε } 
goal: (𝓝 a).HasBasis (fun ε => 0 < ε) fun ε => {b | |b - a| < ε}

full_name: nhds_basis_Ioo_pos
code: theorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :
    (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) 
goal: (𝓝 a).HasBasis (fun ε => 0 < ε) fun ε => Set.Ioo (a - ε) (a + ε)

full_name: nhds_basis_Icc_pos
code: theorem nhds_basis_Icc_pos [NoMaxOrder α] [DenselyOrdered α] (a : α) :
    (𝓝 a).HasBasis ((0 : α) < ·) fun ε ↦ Icc (a - ε) (a + ε) 
goal: (𝓝 a).HasBasis (fun x => 0 < x) fun ε => Set.Icc (a - ε) (a + ε)

full_name: nhds_basis_zero_abs_sub_lt
code: theorem nhds_basis_zero_abs_sub_lt [NoMaxOrder α] :
    (𝓝 (0 : α)).HasBasis (fun ε : α => (0 : α) < ε) fun ε => { b | |b| < ε } 
goal: (𝓝 0).HasBasis (fun ε => 0 < ε) fun ε => {b | |b| < ε}

full_name: nhds_basis_Ioo_pos_of_pos
code: /-- If `a` is positive we can form a basis from only nonnegative `Set.Ioo` intervals -/
theorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder α] {a : α} (ha : 0 < a) :
    (𝓝 a).HasBasis (fun ε : α => (0 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a - ε) (a + ε) 
goal: (𝓝 a).HasBasis (fun ε => 0 < ε ∧ ε ≤ a) fun ε => Set.Ioo (a - ε) (a + ε)

full_name: Set.OrdConnected.mem_nhdsWithin_Ici
code: /-- If `S` is order-connected and contains two points `x < y`, then `S` is a right neighbourhood
of `x`. -/
lemma mem_nhdsWithin_Ici [NoMaxOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[≥] x 
goal: S ∈ 𝓝[≥] x

full_name: Set.OrdConnected.mem_nhdsWithin_Ioi
code: /-- If `S` is order-connected and contains two points `x < y`, then `S` is a punctured right
neighbourhood of `x`. -/
lemma mem_nhdsWithin_Ioi [NoMaxOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[>] x 
goal: S ∈ 𝓝[>] x

full_name: Set.OrdConnected.mem_nhdsWithin_Iic
code: /-- If `S` is order-connected and contains two points `x < y`, then `S` is a left neighbourhood
of `y`. -/
lemma mem_nhdsWithin_Iic [NoMinOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[≤] y 
goal: S ∈ 𝓝[≤] y

full_name: Set.OrdConnected.mem_nhdsWithin_Iio
code: /-- If `S` is order-connected and contains two points `x < y`, then `S` is a punctured left
neighbourhood of `y`. -/
lemma mem_nhdsWithin_Iio [NoMinOrder α] {S : Set α} (hS : OrdConnected S)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) :
    S ∈ 𝓝[<] y 
goal: S ∈ 𝓝[<] y