case intro.intro
o : Ordinal
h' : o < type fun x x_1 => x < x_1
h : Ordinal.lift o < ω
⊢ Ordinal.lift o < ord ℵ₀
o : Ordinal
h : o < ω
⊢ o < ord ℵ₀
rcases Ordinal.lt_lift_iff.1 h with ⟨o, rfl, h'⟩
no goals
case intro.intro
o : Ordinal
h' : o < type fun x x_1 => x < x_1
h : Ordinal.lift o < ω
⊢ card (typein (fun x x_1 => x < x_1) (enum (fun x x_1 => x < x_1) o h')) < ℵ₀
exact lt_aleph0_iff_fintype.2 ⟨Set.fintypeLTNat _⟩