simp only [mem_minimals_iff, mem_setOf_eq, and_congr_right_iff]
α : Type u_1
P : Set α → Prop
hP : ∀ ⦃s t : Set α⦄, P s → s ⊆ t → P t
s : Set α
⊢ s ∈ minimals (fun x x_1 => x ⊆ x_1) {t | P t} ↔ P s ∧ ∀ x ∈ s, ¬P (s \ {x})
α : Type u_1
P : Set α → Prop
hP : ∀ ⦃s t : Set α⦄, P s → s ⊆ t → P t
s : Set α
⊢ P s → ((∀ ⦃y : Set α⦄, P y → y ⊆ s → s = y) ↔ ∀ x ∈ s, ¬P (s \ {x}))