intro _ _ hI hI₂
case mp
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
hP : Set.Nonempty {r | Prime r}
⊢ UniqueFactorizationMonoid R → ∀ (I : Ideal R), I ≠ ⊥ → Ideal.IsPrime I → ∃ x ∈ I, Prime x

case mpr
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
hP : Set.Nonempty {r | Prime r}
⊢ (∀ (I : Ideal R), I ≠ ⊥ → Ideal.IsPrime I → ∃ x ∈ I, Prime x) → UniqueFactorizationMonoid R
case mp
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
hP : Set.Nonempty {r | Prime r}
a✝ : UniqueFactorizationMonoid R
I✝ : Ideal R
hI : I✝ ≠ ⊥
hI₂ : Ideal.IsPrime I✝
⊢ ∃ x ∈ I✝, Prime x

case mpr
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
hP : Set.Nonempty {r | Prime r}
⊢ (∀ (I : Ideal R), I ≠ ⊥ → Ideal.IsPrime I → ∃ x ∈ I, Prime x) → UniqueFactorizationMonoid R