Theorem: generateFrom_Ico_mem_le_borel
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: apply generateFrom_le
state_before: α✝ : Type ?u.606888
β : Type ?u.606891
γ : Type ?u.606894
γ₂ : Type ?u.606897
δ : Type ?u.606900
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type ?u.606993
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
⊢ MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} ≤ borel α

tactic: borelize α
state_before: case h
α✝ : Type ?u.606888
β : Type ?u.606891
γ : Type ?u.606894
γ₂ : Type ?u.606897
δ : Type ?u.606900
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type ?u.606993
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
⊢ ∀ (t_1 : Set α), t_1 ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1

tactic: rintro _ ⟨a, -, b, -, -, rfl⟩
state_before: case h
α✝ : Type ?u.606888
β : Type ?u.606891
γ : Type ?u.606894
γ₂ : Type ?u.606897
δ : Type ?u.606900
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type ?u.606993
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
⊢ ∀ (t_1 : Set α), t_1 ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1

tactic: exact measurableSet_Ico
state_before: case h.intro.intro.intro.intro.intro
α✝ : Type ?u.606888
β : Type ?u.606891
γ : Type ?u.606894
γ₂ : Type ?u.606897
δ : Type ?u.606900
ι : Sort y
s✝ t✝ u : Set α✝
inst✝¹⁹ : TopologicalSpace α✝
inst✝¹⁸ : MeasurableSpace α✝
inst✝¹⁷ : OpensMeasurableSpace α✝
inst✝¹⁶ : TopologicalSpace β
inst✝¹⁵ : MeasurableSpace β
inst✝¹⁴ : OpensMeasurableSpace β
inst✝¹³ : TopologicalSpace γ
inst✝¹² : MeasurableSpace γ
inst✝¹¹ : BorelSpace γ
inst✝¹⁰ : TopologicalSpace γ₂
inst✝⁹ : MeasurableSpace γ₂
inst✝⁸ : BorelSpace γ₂
inst✝⁷ : MeasurableSpace δ
α' : Type ?u.606993
inst✝⁶ : TopologicalSpace α'
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : LinearOrder α✝
inst✝³ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderClosedTopology α
s t : Set α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
a b : α
⊢ MeasurableSet (Ico a b)