Theorem: IsClosed.Icc_subset_of_forall_mem_nhdsWithin
File: Mathlib/Topology/Order/IntermediateValue.lean

Proof steps:
state_before: α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
⊢ Icc a b ⊆ s
state_after: α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
⊢ ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty
tactic: apply hs.Icc_subset_of_forall_exists_gt ha
state_before: α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
⊢ ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty
state_after: case intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
x : α
hxs : x ∈ s
hxab : x ∈ Ico a b
y : α
hyxb : y ∈ Ioi x
⊢ (s ∩ Ioc x y).Nonempty
tactic: rintro x ⟨hxs, hxab⟩ y hyxb
state_before: case intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
x : α
hxs : x ∈ s
hxab : x ∈ Ico a b
y : α
hyxb : y ∈ Ioi x
⊢ (s ∩ Ioc x y).Nonempty
state_after: case intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
x : α
hxs : x ∈ s
hxab : x ∈ Ico a b
y : α
hyxb : y ∈ Ioi x
this : s ∩ Ioc x y ∈ 𝓝[>] x
⊢ (s ∩ Ioc x y).Nonempty
tactic: have : s ∩ Ioc x y ∈ 𝓝[>] x :=
    inter_mem (hgt x ⟨hxs, hxab⟩) (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hyxb⟩)
state_before: case intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a✝ b✝ a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x
x : α
hxs : x ∈ s
hxab : x ∈ Ico a b
y : α
hyxb : y ∈ Ioi x
this : s ∩ Ioc x y ∈ 𝓝[>] x
⊢ (s ∩ Ioc x y).Nonempty
state_after: no goals
tactic: exact (nhdsWithin_Ioi_self_neBot' ⟨b, hxab.2⟩).nonempty_of_mem this