File: Mathlib/Topology/Order/NhdsSet.lean

full_name: nhdsSet_Ioi
code: @[simp] theorem nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) 
goal: 𝓝ˢ (Set.Ioi a) = 𝓟 (Set.Ioi a)

full_name: nhdsSet_Iio
code: @[simp] theorem nhdsSet_Iio : 𝓝ˢ (Iio a) = 𝓟 (Iio a) 
goal: 𝓝ˢ (Set.Iio a) = 𝓟 (Set.Iio a)

full_name: nhdsSet_Ioo
code: @[simp] theorem nhdsSet_Ioo : 𝓝ˢ (Ioo a b) = 𝓟 (Ioo a b) 
goal: 𝓝ˢ (Set.Ioo a b) = 𝓟 (Set.Ioo a b)

full_name: nhdsSet_Ici
code: theorem nhdsSet_Ici : 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a) 
goal: 𝓝ˢ (Set.Ici a) = 𝓝 a ⊔ 𝓟 (Set.Ioi a)

full_name: nhdsSet_Iic
code: theorem nhdsSet_Iic : 𝓝ˢ (Iic a) = 𝓝 a ⊔ 𝓟 (Iio a) 
goal: 𝓝ˢ (Set.Iic a) = 𝓝 a ⊔ 𝓟 (Set.Iio a)

full_name: nhdsSet_Ico
code: theorem nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) 
goal: 𝓝ˢ (Set.Ico a b) = 𝓝 a ⊔ 𝓟 (Set.Ioo a b)

full_name: nhdsSet_Ioc
code: theorem nhdsSet_Ioc (h : a < b) : 𝓝ˢ (Ioc a b) = 𝓝 b ⊔ 𝓟 (Ioo a b) 
goal: 𝓝ˢ (Set.Ioc a b) = 𝓝 b ⊔ 𝓟 (Set.Ioo a b)

full_name: nhdsSet_Icc
code: theorem nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) 
goal: 𝓝ˢ (Set.Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Set.Ioo a b)

full_name: Ioi_mem_nhdsSet_Ici_iff
code: @[simp]
theorem Ioi_mem_nhdsSet_Ici_iff : Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b 
goal: Set.Ioi a ∈ 𝓝ˢ (Set.Ici b) ↔ a < b

full_name: Ici_mem_nhdsSet_Ici
code: theorem Ici_mem_nhdsSet_Ici (h : a < b) : Ici a ∈ 𝓝ˢ (Ici b) 
goal: Set.Ici a ∈ 𝓝ˢ (Set.Ici b)

full_name: Iio_mem_nhdsSet_Iic_iff
code: theorem Iio_mem_nhdsSet_Iic_iff : Iio b ∈ 𝓝ˢ (Iic a) ↔ a < b 
goal: Set.Iio b ∈ 𝓝ˢ (Set.Iic a) ↔ a < b

full_name: Iic_mem_nhdsSet_Iic
code: theorem Iic_mem_nhdsSet_Iic (h : a < b) : Iic b ∈ 𝓝ˢ (Iic a) 
goal: Set.Iic b ∈ 𝓝ˢ (Set.Iic a)

full_name: Ioi_mem_nhdsSet_Icc
code: theorem Ioi_mem_nhdsSet_Icc (h : a < b) : Ioi a ∈ 𝓝ˢ (Icc b c) 
goal: Set.Ioi a ∈ 𝓝ˢ (Set.Icc b c)

full_name: Ici_mem_nhdsSet_Icc
code: theorem Ici_mem_nhdsSet_Icc (h : a < b) : Ici a ∈ 𝓝ˢ (Icc b c) 
goal: Set.Ici a ∈ 𝓝ˢ (Set.Icc b c)

full_name: Iio_mem_nhdsSet_Icc
code: theorem Iio_mem_nhdsSet_Icc (h : b < c) : Iio c ∈ 𝓝ˢ (Icc a b) 
goal: Set.Iio c ∈ 𝓝ˢ (Set.Icc a b)

full_name: Iic_mem_nhdsSet_Icc
code: theorem Iic_mem_nhdsSet_Icc (h : b < c) : Iic c ∈ 𝓝ˢ (Icc a b) 
goal: Set.Iic c ∈ 𝓝ˢ (Set.Icc a b)

full_name: Ioo_mem_nhdsSet_Icc
code: theorem Ioo_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Icc b c) 
goal: Set.Ioo a d ∈ 𝓝ˢ (Set.Icc b c)

full_name: Ico_mem_nhdsSet_Icc
code: theorem Ico_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Icc b c) 
goal: Set.Ico a d ∈ 𝓝ˢ (Set.Icc b c)

full_name: Ioc_mem_nhdsSet_Icc
code: theorem Ioc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Icc b c) 
goal: Set.Ioc a d ∈ 𝓝ˢ (Set.Icc b c)

full_name: Icc_mem_nhdsSet_Icc
code: theorem Icc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Icc b c) 
goal: Set.Icc a d ∈ 𝓝ˢ (Set.Icc b c)

full_name: Ici_mem_nhdsSet_Ico
code: theorem Ici_mem_nhdsSet_Ico (h : a < b) : Ici a ∈ 𝓝ˢ (Ico b c) 
goal: Set.Ici a ∈ 𝓝ˢ (Set.Ico b c)

full_name: Ioi_mem_nhdsSet_Ico
code: theorem Ioi_mem_nhdsSet_Ico (h : a < b) : Ioi a ∈ 𝓝ˢ (Ico b c) 
goal: Set.Ioi a ∈ 𝓝ˢ (Set.Ico b c)

full_name: Iio_mem_nhdsSet_Ico
code: theorem Iio_mem_nhdsSet_Ico (h : b ≤ c) : Iio c ∈ 𝓝ˢ (Ico a b) 
goal: Set.Iio c ∈ 𝓝ˢ (Set.Ico a b)

full_name: Iic_mem_nhdsSet_Ico
code: theorem Iic_mem_nhdsSet_Ico (h : b ≤ c) : Iic c ∈ 𝓝ˢ (Ico a b) 
goal: Set.Iic c ∈ 𝓝ˢ (Set.Ico a b)

full_name: Ioo_mem_nhdsSet_Ico
code: theorem Ioo_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioo a d ∈ 𝓝ˢ (Ico b c) 
goal: Set.Ioo a d ∈ 𝓝ˢ (Set.Ico b c)

full_name: Icc_mem_nhdsSet_Ico
code: theorem Icc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Icc a d ∈ 𝓝ˢ (Ico b c) 
goal: Set.Icc a d ∈ 𝓝ˢ (Set.Ico b c)

full_name: Ioc_mem_nhdsSet_Ico
code: theorem Ioc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioc a d ∈ 𝓝ˢ (Ico b c) 
goal: Set.Ioc a d ∈ 𝓝ˢ (Set.Ico b c)

full_name: Ico_mem_nhdsSet_Ico
code: theorem Ico_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ico a d ∈ 𝓝ˢ (Ico b c) 
goal: Set.Ico a d ∈ 𝓝ˢ (Set.Ico b c)

full_name: Ioi_mem_nhdsSet_Ioc
code: theorem Ioi_mem_nhdsSet_Ioc (h : a ≤ b) : Ioi a ∈ 𝓝ˢ (Ioc b c) 
goal: Set.Ioi a ∈ 𝓝ˢ (Set.Ioc b c)

full_name: Iio_mem_nhdsSet_Ioc
code: theorem Iio_mem_nhdsSet_Ioc (h : b < c) : Iio c ∈ 𝓝ˢ (Ioc a b) 
goal: Set.Iio c ∈ 𝓝ˢ (Set.Ioc a b)

full_name: Ici_mem_nhdsSet_Ioc
code: theorem Ici_mem_nhdsSet_Ioc (h : a ≤ b) : Ici a ∈ 𝓝ˢ (Ioc b c) 
goal: Set.Ici a ∈ 𝓝ˢ (Set.Ioc b c)

full_name: Iic_mem_nhdsSet_Ioc
code: theorem Iic_mem_nhdsSet_Ioc (h : b < c) : Iic c ∈ 𝓝ˢ (Ioc a b) 
goal: Set.Iic c ∈ 𝓝ˢ (Set.Ioc a b)

full_name: Ioo_mem_nhdsSet_Ioc
code: theorem Ioo_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Ioc b c) 
goal: Set.Ioo a d ∈ 𝓝ˢ (Set.Ioc b c)

full_name: Icc_mem_nhdsSet_Ioc
code: theorem Icc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Ioc b c) 
goal: Set.Icc a d ∈ 𝓝ˢ (Set.Ioc b c)

full_name: Ioc_mem_nhdsSet_Ioc
code: theorem Ioc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Ioc b c) 
goal: Set.Ioc a d ∈ 𝓝ˢ (Set.Ioc b c)

full_name: Ico_mem_nhdsSet_Ioc
code: theorem Ico_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Ioc b c) 
goal: Set.Ico a d ∈ 𝓝ˢ (Set.Ioc b c)

full_name: hasBasis_nhdsSet_Iic_Iio
code: theorem hasBasis_nhdsSet_Iic_Iio (a : α) [h : Nonempty (Ioi a)] :
    HasBasis (𝓝ˢ (Iic a)) (a < ·) Iio 
goal: (𝓝ˢ (Set.Iic a)).HasBasis (fun x => a < x) Set.Iio

full_name: hasBasis_nhdsSet_Iic_Iic
code: theorem hasBasis_nhdsSet_Iic_Iic (a : α) [NeBot (𝓝[>] a)] :
    HasBasis (𝓝ˢ (Iic a)) (a < ·) Iic 
goal: (𝓝ˢ (Set.Iic a)).HasBasis (fun x => a < x) Set.Iic

full_name: Iic_mem_nhdsSet_Iic_iff
code: @[simp]
theorem Iic_mem_nhdsSet_Iic_iff {a b : α} [NeBot (𝓝[>] b)] : Iic a ∈ 𝓝ˢ (Iic b) ↔ b < a 
goal: Set.Iic a ∈ 𝓝ˢ (Set.Iic b) ↔ b < a

full_name: hasBasis_nhdsSet_Ici_Ioi
code: theorem hasBasis_nhdsSet_Ici_Ioi (a : α) [Nonempty (Iio a)] :
    HasBasis (𝓝ˢ (Ici a)) (· < a) Ioi 
goal: (𝓝ˢ (Set.Ici a)).HasBasis (fun x => x < a) Set.Ioi

full_name: hasBasis_nhdsSet_Ici_Ici
code: theorem hasBasis_nhdsSet_Ici_Ici (a : α) [NeBot (𝓝[<] a)] :
    HasBasis (𝓝ˢ (Ici a)) (· < a) Ici 
goal: (𝓝ˢ (Set.Ici a)).HasBasis (fun x => x < a) Set.Ici

full_name: Ici_mem_nhdsSet_Ici_iff
code: @[simp]
theorem Ici_mem_nhdsSet_Ici_iff {a b : α} [NeBot (𝓝[<] b)] : Ici a ∈ 𝓝ˢ (Ici b) ↔ a < b 
goal: Set.Ici a ∈ 𝓝ˢ (Set.Ici b) ↔ a < b