simp_rw [cocircuit_def, circuit_def, ← Set.ext_iff, dep_iff, ← coindep_def, dual_ground,
  coindep_iff_exists', not_and, imp_not_comm (b := (_ ⊆ _)), not_exists, not_and, subset_diff,
  not_and, not_disjoint_iff_nonempty_inter]
α : Type u_1
M : Matroid α
C C' I X C₁ C₂ R : Set α
e f x y : α
B : Set α
⊢ ∀ {K : Set α}, M.Cocircuit K ↔ K ∈ minimals (fun x x_1 => x ⊆ x_1) {X | ∀ (B : Set α), M.Base B → (X ∩ B).Nonempty}
α : Type u_1
M : Matroid α
C C' I X C₁ C₂ R : Set α
e f x y : α
B : Set α
⊢ minimals (fun x x_1 => x ⊆ x_1) {X | (X ⊆ M.E → ∀ (x : Set α), M.Base x → x ⊆ M.E → (x ∩ X).Nonempty) ∧ X ⊆ M.E} =
    minimals (fun x x_1 => x ⊆ x_1) {X | ∀ (B : Set α), M.Base B → (X ∩ B).Nonempty}