have : Set.insert x I \ {x} = I :=
  Set.insert_diff_self_of_not_mem (Set.not_mem_of_mem_diff hx)
case refine_1
α : Type u_1
A B I J X : Set α
x : α
M : RankMatroid α
hI_indep : M.Indep I
hx : x ∈ M.E \ I
hIx_indep : M.Indep (Set.insert x I)
h : M.relRank (Set.insert x I \ {x}) (Set.insert x I) > 0
⊢ M.relRank I (Set.insert x I) > 0
case refine_1
α : Type u_1
A B I J X : Set α
x : α
M : RankMatroid α
hI_indep : M.Indep I
hx : x ∈ M.E \ I
hIx_indep : M.Indep (Set.insert x I)
h : M.relRank (Set.insert x I \ {x}) (Set.insert x I) > 0
this : Set.insert x I \ {x} = I
⊢ M.relRank I (Set.insert x I) > 0