Theorem: TFAE_mem_nhdsWithin_Ici
File: Mathlib/Topology/Order/LeftRightNhds.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
⊢ [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].TFAE
state_after: case tfae_1_iff_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
⊢ s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a

α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
⊢ [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].TFAE
tactic: tfae_have 1 ↔ 2
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
⊢ [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].TFAE
state_after: case tfae_1_iff_3
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
⊢ s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a

α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
⊢ [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].TFAE
tactic: tfae_have 1 ↔ 3
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
⊢ [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].TFAE
state_after: case tfae_1_iff_5
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
⊢ s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s

α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
⊢ [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].TFAE
tactic: tfae_have 1 ↔ 5
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
⊢ [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].TFAE
state_after: case tfae_4_to_5
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
⊢ (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s

α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s
⊢ [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].TFAE
tactic: tfae_have 4 → 5
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s
⊢ [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].TFAE
state_after: case tfae_5_to_4
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s
⊢ (∃ u ∈ Ioi a, Ico a u ⊆ s) → ∃ u ∈ Ioc a b, Ico a u ⊆ s

α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_5_to_4 : (∃ u ∈ Ioi a, Ico a u ⊆ s) → ∃ u ∈ Ioc a b, Ico a u ⊆ s
⊢ [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].TFAE
tactic: tfae_have 5 → 4
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_5_to_4 : (∃ u ∈ Ioi a, Ico a u ⊆ s) → ∃ u ∈ Ioc a b, Ico a u ⊆ s
⊢ [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].TFAE
state_after: no goals
tactic: tfae_finish