. apply Set.ext; intros x
  rw [Set.mem_iff, Set.mem_union, Set.mem_compl_iff, imp_iff_not_or]
case heq
α : Type u_1
f : Filter α
s t : Set α
h : {x | x ∈ t → x ∈ s} ∈ f
⊢ {x | x ∈ t → x ∈ s} = tᶜ ∪ s

case mpr
α : Type u_1
f : Filter α
s t : Set α
h : {x | x ∈ t → x ∈ s} ∈ f
heq : {x | x ∈ t → x ∈ s} = tᶜ ∪ s
⊢ s ∈ f ⊓ 𝓟 t
case mpr
α : Type u_1
f : Filter α
s t : Set α
h : {x | x ∈ t → x ∈ s} ∈ f
heq : {x | x ∈ t → x ∈ s} = tᶜ ∪ s
⊢ s ∈ f ⊓ 𝓟 t