have hssE : Js i₀ ∪ (⋃ i ∈ ({i₀}ᶜ : Set ι), Ks i) ⊆ M.E := by
  refine union_subset (hJs i₀).1.indep.subset_ground ?_
  simp only [mem_compl_iff, mem_singleton_iff, iUnion_subset_iff]
  exact fun i _ ↦ (huKs i).indep.subset_ground
case intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B I J✝ X X' Y Y' F : Set α
Xs Ys Js✝ Is : ι → Set α
i j : ι
e✝ f : α
h : M.SkewFamily Xs
hIX : ∀ (i : ι), Is i ⊆ Xs i
hIs : ∀ (i : ι), M.Indep (Is i)
Js : ι → Set α
hJs : ∀ (i : ι), M.Basis (Js i) (Xs i) ∧ Is i ⊆ Js i
J : Set α
hJ : M.Basis J (⋃ i, Js i)
hcon : ¬M.Indep (⋃ i, Js i)
i₀ : ι
e : α
hei₀ : e ∈ Js i₀
heJ : e ∉ J
Ks : ι → Set α
hdj : Pairwise (Disjoint on Ks)
hKs : M.Basis (⋃ i, Ks i) (⋃ i, Xs i)
huKs : ∀ (i : ι), M.Basis (Ks i) (Xs i)
⊢ False
case intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B I J✝ X X' Y Y' F : Set α
Xs Ys Js✝ Is : ι → Set α
i j : ι
e✝ f : α
h : M.SkewFamily Xs
hIX : ∀ (i : ι), Is i ⊆ Xs i
hIs : ∀ (i : ι), M.Indep (Is i)
Js : ι → Set α
hJs : ∀ (i : ι), M.Basis (Js i) (Xs i) ∧ Is i ⊆ Js i
J : Set α
hJ : M.Basis J (⋃ i, Js i)
hcon : ¬M.Indep (⋃ i, Js i)
i₀ : ι
e : α
hei₀ : e ∈ Js i₀
heJ : e ∉ J
Ks : ι → Set α
hdj : Pairwise (Disjoint on Ks)
hKs : M.Basis (⋃ i, Ks i) (⋃ i, Xs i)
huKs : ∀ (i : ι), M.Basis (Ks i) (Xs i)
hssE : Js i₀ ∪ ⋃ i ∈ {i₀}ᶜ, Ks i ⊆ M.E
⊢ False