have : 𝚺₁-Predicate fun x ↦ ∃ y, ⟪x, y⟫ ∈ s :=
  DefinablePred.of_iff (fun x ↦ ∃ y < s, ⟪x, y⟫ ∈ s)
    (fun x ↦ ⟨by rintro ⟨y, hy⟩; exact ⟨y, lt_of_le_of_lt (le_pair_right x y) (lt_of_mem hy), hy⟩,
              by rintro ⟨y, _, hy⟩; exact ⟨y, hy⟩⟩)
    (by definability)
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
⊢ ∃! d, ∀ (x : M), x ∈ d ↔ ∃ y, ⟪x, y⟫ ∈ s
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
s : M
this : 𝚺₁-Predicate fun x => ∃ y, ⟪x, y⟫ ∈ s
⊢ ∃! d, ∀ (x : M), x ∈ d ↔ ∃ y, ⟪x, y⟫ ∈ s