-- Module: Mathlib.Algebra.Module.Submodule.Union
Goal:
ι : Type u_1
K : Type u_2
M : Type u_3
inst : Field K
inst_1 : AddCommGroup M
inst_2 : Module K M
s : Finset ι
p : ι → Submodule K M
h₁ : ∀ (i : ι), p i ≠ ⊤
h₂ : (↑s.card : ℕ∞) < ENat.card K
empty :
  ∀ (h₂ : (↑(Finset.card.{u_1} (α := ι) ∅) : ℕ∞) < ENat.card K),
    SSubset.{u_3} (α := Set M) (⋃ (i : ι), ⋃ (_ : Membership.mem (γ := Finset ι) ∅ i), (↑(p i) : Set M)) univ
insert :
  ∀ (j : ι) (s : Finset ι) (hj : j ∉ s)
    (hj' : ∀ (h₂ : (↑s.card : ℕ∞) < ENat.card K), SSubset.{u_3} (α := Set M) (⋃ i ∈ s, (↑(p i) : Set M)) univ)
    (h₂ : (↑(Insert.insert j s).card : ℕ∞) < ENat.card K),
    SSubset.{u_3} (α := Set M) (⋃ i ∈ Insert.insert j s, (↑(p i) : Set M)) univ
⊢ SSubset.{u_3} (α := Set M) (⋃ i ∈ s, (↑(p i) : Set M)) univ

Tactic:
classical
  induction s using Finset.induction_on with
  | empty => simp
  | insert j s hj hj' =>
    simp only [ssubset_univ_iff] at hj' ⊢
    rcases s.eq_empty_or_nonempty with rfl | hs
    · simpa using h₁ j
    replace h₂ : s.card + 1 < ENat.card K := by simpa [Finset.card_insert_of_notMem hj] using h₂
    specialize hj' (lt_trans ENat.natCast_lt_succ h₂)
    contrapose hj'
    replace hj' : (p j : Set M) ∪ (⋃ i ∈ s, p i) = univ := by
      simpa only [Finset.mem_insert, iUnion_iUnion_eq_or_left] using hj'
    suffices (p j : Set M) ⊆ ⋃ i ∈ s, p i by rwa [union_eq_right.mpr this] at hj'
    intro x (hx : x ∈ p j)
    rcases eq_or_ne x 0 with rfl | hx₀
    · simpa using hs
    obtain ⟨y, hy⟩ : ∃ y, y ∉ p j := by specialize h₁ j; contrapose! h₁; ext; simp [h₁]
    have hy₀ : y ≠ 0 := by aesop
    let sxy := {x + t • y | (t : K) (ht : t ≠ 0)}
    have hsxy : sxy ⊆ ⋃ i ∈ s, p i := by
      suffices Disjoint sxy (p j) from this.subset_right_of_subset_union <| hj' ▸ sxy.subset_univ
      rw [Set.disjoint_iff]
      rintro - ⟨⟨t, ht₀, rfl⟩, ht : x + t • y ∈ p j⟩
      rw [(p j).add_mem_iff_right hx, (p j).smul_mem_iff ht₀] at ht
      contradiction
    obtain ⟨k, hk, t₁, t₂, ht, ht₁, ht₂⟩ : ∃ᵉ (k ∈ s) (t₁ : K) (t₂ : K),
        t₁ ≠ t₂ ∧ x + t₁ • y ∈ p k ∧ x + t₂ • y ∈ p k := by
      suffices ∃ᵉ (k ∈ s) (z₁ ∈ sxy) (z₂ ∈ sxy), z₁ ≠ z₂ ∧ z₁ ∈ p k ∧ z₂ ∈ p k by
        obtain ⟨k, hk, -, ⟨t₁, -, rfl⟩, -, ⟨t₂, -, rfl⟩, htne, ht₁, ht₂⟩ := this
        exact ⟨k, hk, t₁, t₂, by aesop, ht₁, ht₂⟩
      choose f hf using fun z : sxy ↦ mem_iUnion.mp (hsxy z.property)
      have hf' : MapsTo f univ s := fun z _ ↦ by specialize hf z; aesop
      suffices ∃ z₁ z₂, z₁ ≠ z₂ ∧ f z₁ = f z₂ by
        obtain ⟨z₁, z₂, hne, heq⟩ := this
        exact ⟨f z₁, hf' (mem_univ _), z₁, z₁.property, z₂, z₂.property,
          Subtype.coe_ne_coe.mpr hne, by specialize hf z₁; simp_all, by specialize hf z₂; aesop⟩
      have key : s.card < sxy.encard := by
        refine lt_of_add_lt_add_right <| lt_of_lt_of_le h₂ ?_
        have : Injective (fun t : K ↦ x + t • y) :=
          fun t₁ t₂ ht ↦ smul_left_injective K hy₀ <| by simpa using ht
        have aux : sxy = ((fun t : K ↦ x + t • y) '' {t | t ≠ 0}) := by ext; simp [sxy]
        rw [aux, this.encard_image, encard_ne_add_one]
      obtain ⟨z₁, -, z₂, -, h⟩ := exists_ne_map_eq_of_encard_lt_of_maps_to (by simpa) hf'
      exact ⟨z₁, z₂, h⟩
    replace ht : y ∈ p k := by
      have : (t₁ - t₂) • y ∈ p k := by convert! sub_mem ht₁ ht₂ using 1; module
      refine ((p k).smul_mem_iff ?_).mp this
      rwa [sub_ne_zero]
    replace ht : x ∈ p k := by convert! sub_mem ht₁ ((p k).smul_mem t₁ ht); simp
    simpa using ⟨k, hk, ht⟩

Premises used: Insert.insert, Set.ssubset_univ_iff, rfl, Finset.eq_empty_or_nonempty