-- Mathlib/Topology/Order/DenselyOrdered.lean
theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s]
    (hs : Dense s) :
    ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t

Proof state:
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
⊢ ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t

Tactic:
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩

Result:
case intro.intro.intro
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t

Proof state:
case intro.intro.intro.refine_1
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ t \ ({x | IsBot x} ∪ {x | IsTop x}) ⊆ s

Tactic:
exact diff_subset.trans hts

Result:
no goals

Proof state:
case intro.intro.intro.refine_5
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
x : α
hx : IsTop x
⊢ x ∉ t \ ({x | IsBot x} ∪ {x | IsTop x})

Tactic:
simp [hx]

Result:
no goals

Proof state:
case intro.intro.intro.refine_3
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ Dense (t \ ({x | IsBot x} ∪ {x | IsTop x}))

Tactic:
exact htd.diff_finite ((subsingleton_isBot α).finite.union (subsingleton_isTop α).finite)

Result:
no goals

Proof state:
case intro.intro.intro.refine_4
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
x : α
hx : IsBot x
⊢ x ∉ t \ ({x | IsBot x} ∪ {x | IsTop x})

Tactic:
simp [hx]

Result:
no goals

Proof state:
case intro.intro.intro.refine_2
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ (t \ ({x | IsBot x} ∪ {x | IsTop x})).Countable

Tactic:
exact htc.mono diff_subset

Result:
no goals

Proof state:
case intro.intro.intro
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ (x : α), IsBot x → x ∉ t) ∧ ∀ (x : α), IsTop x → x ∉ t

Tactic:
refine ⟨t \ ({ x | IsBot x } ∪ { x | IsTop x }), ?_, ?_, ?_, fun x hx => ?_, fun x hx => ?_⟩

Result:
case intro.intro.intro.refine_1
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ t \ ({x | IsBot x} ∪ {x | IsTop x}) ⊆ s

case intro.intro.intro.refine_2
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ (t \ ({x | IsBot x} ∪ {x | IsTop x})).Countable

case intro.intro.intro.refine_3
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
⊢ Dense (t \ ({x | IsBot x} ∪ {x | IsTop x}))

case intro.intro.intro.refine_4
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
x : α
hx : IsBot x
⊢ x ∉ t \ ({x | IsBot x} ∪ {x | IsTop x})

case intro.intro.intro.refine_5
α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
inst✝² : DenselyOrdered α
inst✝¹ : Nontrivial α
s : Set α
inst✝ : SeparableSpace ↑s
hs : Dense s
t : Set α
hts : t ⊆ s
htc : t.Countable
htd : Dense t
x : α
hx : IsTop x
⊢ x ∉ t \ ({x | IsBot x} ∪ {x | IsTop x})