α : Type u
s : Set α
⊢ IsOpen ({u | s ∈ u}ᶜ)
α : Type u
s : Set α
⊢ IsClosed {u | s ∈ u}
rw [← isOpen_compl_iff]
case h.e'_3
α : Type u
s : Set α
⊢ {u | s ∈ u}ᶜ = {u | sᶜ ∈ u}
α : Type u
s : Set α
⊢ IsOpen ({u | s ∈ u}ᶜ)
convert ultrafilter_isOpen_basic (sᶜ) using 1
case h.e'_3.h
α : Type u
s : Set α
u : Ultrafilter α
⊢ u ∈ {u | s ∈ u}ᶜ ↔ u ∈ {u | sᶜ ∈ u}
case h.e'_3
α : Type u
s : Set α
⊢ {u | s ∈ u}ᶜ = {u | sᶜ ∈ u}
ext u
no goals
case h.e'_3.h
α : Type u
s : Set α
u : Ultrafilter α
⊢ u ∈ {u | s ∈ u}ᶜ ↔ u ∈ {u | sᶜ ∈ u}
exact Ultrafilter.compl_mem_iff_not_mem.symm