/-- In a semiring of sets `C`, for all set `s ∈ C` and finite set of sets `I ⊆ C`, there is a
finite set of sets in `C` whose union is `s \ ⋃₀ I`.
See `IsSetSemiring.diffFinset₀` for a definition that gives such a set. -/
lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    ∃ (J : Finset (Set α)) (_h_ss : ↑J ⊆ C) (_h_dis : PairwiseDisjoint (J : Set (Set α)) id),
      s \ ⋃₀ I = ⋃₀ J := by
  classical
  revert hI
  refine Finset.induction ?_ ?_ I
  · intro
    simp only [coe_empty, sUnion_empty, diff_empty, exists_prop]
    refine ⟨{s}, singleton_subset_set_iff.mpr hs, ?_⟩
    simp only [coe_singleton, pairwiseDisjoint_singleton, sUnion_singleton, eq_self_iff_true,
      and_self_iff]
  intro t I' _ h h_insert_subset
  rw [coe_insert] at h_insert_subset
  have ht : t ∈ C := h_insert_subset (Set.mem_insert _ _)
  obtain ⟨J, h_ss, h_dis, h_eq⟩ := h ((Set.subset_insert _ _).trans h_insert_subset)
  let Ju : ∀ u ∈ C, Finset (Set α) := fun u hu ↦ hC.diffFinset hu ht
  have hJu_subset : ∀ (u) (hu : u ∈ C), ↑(Ju u hu) ⊆ C := by
    intro u hu x hx
    exact hC.diffFinset_subset hu ht hx
  have hJu_disj : ∀ (u) (hu : u ∈ C), (Ju u hu : Set (Set α)).PairwiseDisjoint id := fun u hu ↦
    hC.pairwiseDisjoint_diffFinset hu ht
  have hJu_sUnion : ∀ (u) (hu : u ∈ C), ⋃₀ (Ju u hu : Set (Set α)) = u \ t :=
    fun u hu ↦ hC.sUnion_diffFinset hu ht
  have hJu_disj' : ∀ (u) (hu : u ∈ C) (v) (hv : v ∈ C) (_h_dis : Disjoint u v),
      Disjoint (⋃₀ (Ju u hu : Set (Set α))) (⋃₀ ↑(Ju v hv)) :=by
    intro u hu v hv huv_disj
    rw [hJu_sUnion, hJu_sUnion]
    exact disjoint_of_subset (Set.diff_subset u t) (Set.diff_subset v t) huv_disj
  let J' : Finset (Set α) := Finset.biUnion (Finset.univ : Finset J) fun u ↦ Ju u (h_ss u.prop)
  have hJ'_subset : ↑J' ⊆ C := by
    intro u
    simp only [Subtype.coe_mk, univ_eq_attach, coe_biUnion, mem_coe, mem_attach, iUnion_true,
      mem_iUnion, Finset.exists_coe, bex_imp]
    intro v hv huvt
    exact hJu_subset v (h_ss hv) huvt
  refine ⟨J', hJ'_subset, ?_, ?_⟩
  · rw [Finset.coe_biUnion]
    refine PairwiseDisjoint.biUnion ?_ ?_
    · simp only [univ_eq_attach, mem_coe, id.def, iSup_eq_iUnion]
      simp_rw [PairwiseDisjoint, Set.Pairwise, Function.onFun]
      intro x _ y _ hxy
      have hxy_disj : Disjoint (x : Set α) y := by
        by_contra h_contra
        refine hxy ?_
        refine Subtype.ext ?_
        exact h_dis.elim x.prop y.prop h_contra
      convert hJu_disj' (x : Set α) (h_ss x.prop) y (h_ss y.prop) hxy_disj
      · rw [sUnion_eq_biUnion]
        congr
      · rw [sUnion_eq_biUnion]
        congr
    · exact fun u _ ↦ hJu_disj _ _
  · rw [coe_insert, sUnion_insert, Set.union_comm, ← Set.diff_diff, h_eq]
    

Proof state:
case refine_2.intro.intro.intro.refine_2
α : Type u_1
C : Set (Set α)
s t✝ : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
t : Set α
I' : Finset (Set α)
a✝ : t ∉ I'
h : ↑I' ⊆ C → ∃ J, ∃ (_ : ↑J ⊆ C) (_ : PairwiseDisjoint (↑J) id), s \ ⋃₀ ↑I' = ⋃₀ ↑J
h_insert_subset : insert t ↑I' ⊆ C
ht : t ∈ C
J : Finset (Set α)
h_ss : ↑J ⊆ C
h_dis : PairwiseDisjoint (↑J) id
h_eq : s \ ⋃₀ ↑I' = ⋃₀ ↑J
Ju : (u : Set α) → u ∈ C → Finset (Set α) := fun u hu => diffFinset hC hu ht
hJu_subset : ∀ (u : Set α) (hu : u ∈ C), ↑(Ju u hu) ⊆ C
hJu_disj : ∀ (u : Set α) (hu : u ∈ C), PairwiseDisjoint (↑(Ju u hu)) id
hJu_sUnion : ∀ (u : Set α) (hu : u ∈ C), ⋃₀ ↑(Ju u hu) = u \ t
hJu_disj' : ∀ (u : Set α) (hu : u ∈ C) (v : Set α) (hv : v ∈ C), Disjoint u v → Disjoint (⋃₀ ↑(Ju u hu)) (⋃₀ ↑(Ju v hv))
J' : Finset (Set α) := Finset.biUnion Finset.univ fun u => Ju ↑u (_ : ↑u ∈ C)
hJ'_subset : ↑J' ⊆ C
⊢ ⋃₀ ↑J \ t = ⋃₀ ↑J'

Next tactic:
simp_rw [sUnion_eq_biUnion, Set.iUnion_diff]