have h_inter : ⋂ e, Is e = J := by
  rw [hIs, ← biInter_eq_iInter (t := fun x _ ↦ I \ {x}), biInter_diff_singleton_eq_diff _ hne,
    diff_diff_right, diff_self, empty_union, inter_eq_self_of_subset_right hJI.subset]
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hI : M.Indep I
hJI : J ⊂ I
h : ∀ e ∈ I \ J, M.cl (I \ {e}) ∈ U
Is : ↑(I \ J) → Set α := fun e => I \ {↑e}
hIs : Is = fun e => I \ {↑e}
hmod : M.ModularFamily Is
hne : (I \ J).Nonempty
⊢ M.cl J ∈ U
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hI : M.Indep I
hJI : J ⊂ I
h : ∀ e ∈ I \ J, M.cl (I \ {e}) ∈ U
Is : ↑(I \ J) → Set α := fun e => I \ {↑e}
hIs : Is = fun e => I \ {↑e}
hmod : M.ModularFamily Is
hne : (I \ J).Nonempty
h_inter : ⋂ e, Is e = J
⊢ M.cl J ∈ U