α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
⊢ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
⊢ QuasiSober α
rw [quasiSober_iff]
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
⊢ ∃ x, IsGenericPoint x t
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
⊢ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S
intro t h h'
case intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
⊢ ∃ x, IsGenericPoint x t
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
⊢ ∃ x, IsGenericPoint x t
obtain ⟨x, hx⟩ := h.1
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
⊢ ∃ x, IsGenericPoint x t
case intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
⊢ ∃ x, IsGenericPoint x t
obtain ⟨U, hU, hU'⟩ : x ∈ ⋃₀ S := by
  rw [hS'']
  trivial
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
⊢ ∃ x, IsGenericPoint x t
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
⊢ ∃ x, IsGenericPoint x t
haveI : QuasiSober U := hS' ⟨U, hU⟩
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsPreirreducible (Subtype.val ⁻¹' t)
⊢ ∃ x, IsGenericPoint x t
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
⊢ ∃ x, IsGenericPoint x t
have H : IsPreirreducible ((↑) ⁻¹' t : Set U) :=
  h.2.preimage (hS ⟨U, hU⟩).openEmbedding_subtype_val
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ ∃ x, IsGenericPoint x t
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsPreirreducible (Subtype.val ⁻¹' t)
⊢ ∃ x, IsGenericPoint x t
replace H : IsIrreducible ((↑) ⁻¹' t : Set U) := ⟨⟨⟨x, hU'⟩, by simpa using hx⟩, H⟩
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ ∃ x, IsGenericPoint x t
use H.genericPoint
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' closure t
⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t
have := continuous_subtype_val.closure_preimage_subset _ H.genericPoint_spec.mem
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' closure t
⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t
rw [h'.closure_eq] at this
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ closure {↑(IsIrreducible.genericPoint H)} ≤ t

case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ t ≤ closure {↑(IsIrreducible.genericPoint H)}
case intro.intro.intro
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t
apply le_antisymm
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ t ≤ closure (Subtype.val '' closure (Subtype.val ⁻¹' t))
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ t ≤ closure {↑(IsIrreducible.genericPoint H)}
rw [← image_singleton, ← closure_image_closure continuous_subtype_val, H.genericPoint_spec.def]
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ t ∩ ↑{ val := U, property := hU } ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ t ≤ closure (Subtype.val '' closure (Subtype.val ⁻¹' t))
refine' (subset_closure_inter_of_isPreirreducible_of_isOpen h.2 (hS ⟨U, hU⟩) ⟨x, hx, hU'⟩).trans
  (closure_mono _)
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ Subtype.val '' (Subtype.val ⁻¹' t) ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ t ∩ ↑{ val := U, property := hU } ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)
rw [← Subtype.image_preimage_coe]
no goals
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ Subtype.val '' (Subtype.val ⁻¹' t) ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)
exact Set.image_subset _ subset_closure
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
⊢ x ∈ ⊤
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
⊢ x ∈ ⋃₀ S
rw [hS'']
no goals
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
⊢ x ∈ ⊤
trivial
no goals
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this : QuasiSober ↑U
H : IsPreirreducible (Subtype.val ⁻¹' t)
⊢ { val := x, property := hU' } ∈ Subtype.val ⁻¹' t
simpa using hx
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ {↑(IsIrreducible.genericPoint H)} ⊆ t
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ closure {↑(IsIrreducible.genericPoint H)} ≤ t
apply h'.closure_subset_iff.mpr
no goals
case intro.intro.intro.a
α : Type u_1
β : Type ?u.12585
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
S : Set (Set α)
hS : ∀ (s : ↑S), IsOpen ↑s
hS' : ∀ (s : ↑S), QuasiSober ↑↑s
hS'' : ⋃₀ S = ⊤
t : Set α
h : IsIrreducible t
h' : IsClosed t
x : α
hx : x ∈ t
U : Set α
hU : U ∈ S
hU' : x ∈ U
this✝ : QuasiSober ↑U
H : IsIrreducible (Subtype.val ⁻¹' t)
this : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t
⊢ {↑(IsIrreducible.genericPoint H)} ⊆ t
simpa using this