simp only [Set.empty_subset, true_and, Set.mem_setOf_eq, and_imp] at hI
case intro
α : Type u_1
I✝ : Set α
M : Matroid α
A B : Set α
hA : A ⊆ B
hB : B ⊆ M.E
I : Set α
hI :
  I ∈
    {a |
      a ∈ {Y | M.Indep Y ∧ ∅ ⊆ Y ∧ Y ⊆ A} ∧
        ∀ ⦃b : Set α⦄, b ∈ {Y | M.Indep Y ∧ ∅ ⊆ Y ∧ Y ⊆ A} → (fun x x_1 => x ⊆ x_1) a b → (fun x x_1 => x ⊆ x_1) b a}
⊢ ∃ I J, I ⊆ J ∧ M.Basis I A ∧ M.Basis J B ∧ M.relRank A B = (J \ I).encard
case intro
α : Type u_1
I✝ : Set α
M : Matroid α
A B : Set α
hA : A ⊆ B
hB : B ⊆ M.E
I : Set α
hI : (M.Indep I ∧ I ⊆ A) ∧ ∀ ⦃b : Set α⦄, M.Indep b → b ⊆ A → I ⊆ b → b ⊆ I
⊢ ∃ I J, I ⊆ J ∧ M.Basis I A ∧ M.Basis J B ∧ M.relRank A B = (J \ I).encard