have heEI : e ∈ M.E \ I := by
  refine' ⟨M.cl_subset_ground _ (he _ hne.some_mem), fun heI ↦ he' _⟩
  refine' mem_cl_of_mem _ (fun X hX' ↦ _) hiI.subset_ground
  rw [← hI.cl_inter_eq_self_of_subset (hIs X hX')]
  exact ⟨he X hX', heI⟩
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e✝ f x y : α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.cl i
he' : e ∉ M.cl (⋂₀ Js)
⊢ False
α : Type u_1
ι : Type u_2
M : Matroid α
F I J X Y B C R : Set α
e✝ f x y : α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.cl i
he' : e ∉ M.cl (⋂₀ Js)
heEI : e ∈ M.E \ I
⊢ False