l3lab/ntp-mathlib huggingface dataset:
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        

Proof state:
case pos.intro.intro.intro
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ ∃ ub ∈ opens, ∀ s ∈ c, s ⊆ ub

Next tactic:
use ⋃₀ c
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        

Proof state:
case h
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ ⋃₀ c ∈ opens ∧ ∀ s ∈ c, s ⊆ ⋃₀ c

Next tactic:
refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · 

Proof state:
case h.refine'_1
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ Sᶜ ⊆ ⋃₀ c

Next tactic:
exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · 

Proof state:
case h.refine'_2
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ IsOpen (⋃₀ c)

Next tactic:
exact isOpen_sUnion fun _ h => (hc h).2.1
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · 

Proof state:
case h.refine'_3
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ Set.Nonempty (⋃₀ c)ᶜ

Next tactic:
convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · 

Proof state:
case h.e'_2
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ = ⋂ U, (↑U)ᶜ

Next tactic:
ext
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            

Proof state:
case h.e'_2.h
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
x✝ : X
⊢ x✝ ∈ (⋃₀ c)ᶜ ↔ x✝ ∈ ⋂ U, (↑U)ᶜ

Next tactic:
simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          

Proof state:
case h.refine'_3
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ Set.Nonempty (⋂ U, (↑U)ᶜ)

Next tactic:
apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · 

Proof state:
case h.refine'_3.htd
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => (↑i)ᶜ

Next tactic:
rintro ⟨U, hU⟩ ⟨U', hU'⟩
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            

Proof state:
case h.refine'_3.htd.mk.mk
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
U : Set X
hU : U ∈ c
U' : Set X
hU' : U' ∈ c
⊢ ∃ z,
    (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U, property := hU }) ((fun i => (↑i)ᶜ) z) ∧
      (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U', property := hU' }) ((fun i => (↑i)ᶜ) z)

Next tactic:
obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
            

Proof state:
case h.refine'_3.htd.mk.mk.intro.intro.intro
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
U : Set X
hU : U ∈ c
U' : Set X
hU' : U' ∈ c
V : Set X
hVc : V ∈ c
hVU : U ⊆ V
hVU' : U' ⊆ V
⊢ ∃ z,
    (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U, property := hU }) ((fun i => (↑i)ᶜ) z) ∧
      (fun x x_1 => x ⊇ x_1) ((fun i => (↑i)ᶜ) { val := U', property := hU' }) ((fun i => (↑i)ᶜ) z)

Next tactic:
exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
            exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩
          · 

Proof state:
case h.refine'_3.htn
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ ∀ (i : { U // U ∈ c }), Set.Nonempty (↑i)ᶜ

Next tactic:
exact fun U => (hc U.2).2.2
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
            exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩
          · exact fun U => (hc U.2).2.2
          · 

Proof state:
case h.refine'_3.htc
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ ∀ (i : { U // U ∈ c }), IsCompact (↑i)ᶜ

Next tactic:
exact fun U => (hc U.2).2.1.isClosed_compl.isCompact
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
            exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩
          · exact fun U => (hc U.2).2.2
          · exact fun U => (hc U.2).2.1.isClosed_compl.isCompact
          · 

Proof state:
case h.refine'_3.htcl
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ ∀ (i : { U // U ∈ c }), IsClosed (↑i)ᶜ

Next tactic:
exact fun U => (hc U.2).2.1.isClosed_compl
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
            exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩
          · exact fun U => (hc U.2).2.2
          · exact fun U => (hc U.2).2.1.isClosed_compl.isCompact
          · exact fun U => (hc U.2).2.1.isClosed_compl
      · 

Proof state:
case neg
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
hcne : ¬Set.Nonempty c
⊢ ∃ ub ∈ opens, ∀ s ∈ c, s ⊆ ub

Next tactic:
use Sᶜ
theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
    (hS : IsClosed S) (hne : S.Nonempty) :
    ∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
      ∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
  let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
  obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ :=
    zorn_subset opens fun c hc hz => by
      by_cases hcne : c.Nonempty
      · obtain ⟨U₀, hU₀⟩ := hcne
        haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
        obtain ⟨U₀compl, -, -⟩ := hc hU₀
        use ⋃₀ c
        refine' ⟨⟨_, _, _⟩, fun U hU _ hx => ⟨U, hU, hx⟩⟩
        · exact fun _ hx => ⟨U₀, hU₀, U₀compl hx⟩
        · exact isOpen_sUnion fun _ h => (hc h).2.1
        · convert_to (⋂ U : { U // U ∈ c }, U.1ᶜ).Nonempty
          · ext
            simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,
              mem_setOf_eq, mem_compl_iff, mem_sUnion]
          apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed
          · rintro ⟨U, hU⟩ ⟨U', hU'⟩
            obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU'
            exact ⟨⟨V, hVc⟩, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'⟩
          · exact fun U => (hc U.2).2.2
          · exact fun U => (hc U.2).2.1.isClosed_compl.isCompact
          · exact fun U => (hc U.2).2.1.isClosed_compl
      · use Sᶜ
        

Proof state:
case h
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : Set.Nonempty S
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Set.Nonempty Uᶜ}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x x_1 => x ⊆ x_1) c
hcne : ¬Set.Nonempty c
⊢ Sᶜ ∈ opens ∧ ∀ s ∈ c, s ⊆ Sᶜ

Next tactic:
refine' ⟨⟨Set.Subset.refl _, isOpen_compl_iff.mpr hS, _⟩, fun U Uc => (hcne ⟨U, Uc⟩).elim⟩