case empty =>
  intros _
  refine ⟨default,?_⟩
  intros h₂ h₃
  contradiction
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑∅ ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑∅ ⊆ fam n
no goals