Theorem: isPreconnected_Icc
File: Mathlib/Topology/Order/IntermediateValue.lean

tactic: rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
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 : α
⊢ ∀ (t t' : Set α),
    IsClosed t →
      IsClosed t' → Icc a b ⊆ t ∪ t' → (Icc a b ∩ t).Nonempty → (Icc a b ∩ t').Nonempty → (Icc a b ∩ (t ∩ t')).Nonempty
state_after: case intro.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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
⊢ (Icc a b ∩ (s ∩ t)).Nonempty

tactic: rcases le_total x y with h | h
state_before: case intro.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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
⊢ (Icc a b ∩ (s ∩ t)).Nonempty
state_after: case intro.intro.inl
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : x ≤ y
⊢ (Icc a b ∩ (s ∩ t)).Nonempty

case intro.intro.inr
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ (Icc a b ∩ (s ∩ t)).Nonempty

tactic: exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
state_before: case intro.intro.inl
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : x ≤ y
⊢ (Icc a b ∩ (s ∩ t)).Nonempty
state_after: no goals

tactic: rw [inter_comm s t]
state_before: case intro.intro.inr
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ (Icc a b ∩ (s ∩ t)).Nonempty
state_after: case intro.intro.inr
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ (Icc a b ∩ (t ∩ s)).Nonempty

tactic: rw [union_comm s t] at hab
state_before: case intro.intro.inr
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ (Icc a b ∩ (t ∩ s)).Nonempty
state_after: case intro.intro.inr
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ t ∪ s
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ (Icc a b ∩ (t ∩ s)).Nonempty

tactic: exact isPreconnected_Icc_aux y x t s h ht hs hab hy hx
state_before: case intro.intro.inr
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 : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ t ∪ s
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ (Icc a b ∩ (t ∩ s)).Nonempty
state_after: no goals