awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
this : s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
rw : 2 * #s = card.{u_1} (α := α) Finset.univ
rw₁ : 2 * #s = #(s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯)
rw₂ : #s + #s = #(s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯)
⊢ 2 * #s = Fintype.card α
have rw₃ : #s + #s = #s + #(map { toFun := compl, inj' := sorry : α ↪ α } s)
Tactic: rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
this : s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
rw : 2 * #s = card.{u_1} (α := α) Finset.univ
rw₁ : 2 * #s = #(s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯)
rw₂ : #s + #s = #(s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯)
rw₃ : #s + #s = #s + #(map { toFun := compl, inj' := ⋯ : α ↪ α } s)
⊢ 2 * #s = Fintype.card α
have rw₄ : #s + #s = #s + #s
Tactic: rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
⊢ 2 * #s = Fintype.card α
have suffices : s.disjUnion (map { toFun := compl, inj' := sorry : α ↪ α } s) sorry = Finset.univ
Tactic: suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
⊢ s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
have rw : (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
Tactic: rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
⊢ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
have refine : ∀ (a : α), a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
Tactic: refine eq_univ_of_forall fun a => ?_
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
a : α
⊢ a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
have simp_rw : a ∈ (↑s : Set α) ∨ aᶜ ∈ (↑s : Set α)
Tactic: simp_rw [mem_union, mem_preimage]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
a : α
⊢ a ∈ (↑s : Set α) ∨ aᶜ ∈ (↑s : Set α)
have by_contra! : ∀ (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)), False
Tactic: by_contra! ha
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
a : α
ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)
⊢ False
have refine : (↑(insert a s) : Set α).Intersecting
Tactic: refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
a : α
ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)
⊢ (↑(insert a s) : Set α).Intersecting
have rw : (insert a (↑s : Set α)).Intersecting
Tactic: rw [coe_insert]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
a : α
ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)
⊢ (insert a (↑s : Set α)).Intersecting
have refine : a ≠ ⊥
Tactic: refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
ha : ⊥ ∉ (↑s : Set α) ∧ ⊥ᶜ ∉ (↑s : Set α)
⊢ (↑{⊤} : Set α).Intersecting
have rw : Intersecting (α := α) {⊤}
Tactic: rw [coe_singleton]
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (hypothesis)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
ha : ⊥ ∉ (↑s : Set α) ∧ ⊥ᶜ ∉ (↑s : Set α)
⊢ False
have this : s ⊆ {⊤} → s = {⊤}
Tactic: have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (hypothesis)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
ha : ⊥ ∉ (↑s : Set α) ∧ ⊥ᶜ ∉ (↑s : Set α)
this : s ⊆ {⊤} → s = {⊤}
⊢ False
have ha₁ : ⊥ ∉ (↑s : Set α) ∧ ⊤ ∉ (↑s : Set α)
Tactic: rw [compl_bot] at ha
Cleared hypotheses: ha, this
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (hypothesis)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t
ha : ⊥ ∉ (↑s : Set α) ∧ ⊤ ∉ (↑s : Set α)
this : s ⊆ {⊤} → s = {⊤}
⊢ False
have this₁ : Subset (α := Finset α) ∅ {⊤} → Eq (α := Finset α) ∅ {⊤}
Tactic: rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₁ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
  s.disjUnion (map { toFun := compl, inj' := sorry : α ↪ α } s) sorry = Finset.univ
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
refine₁ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
    s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₂ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
refine₁ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
    s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
refine₂ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₃ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
refine₁ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
    s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
refine₂ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
refine₃ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₄ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∨ aᶜ ∈ (↑s : Set α)
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
refine₁ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
    s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
refine₂ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
refine₃ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
refine₄ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∨ aᶜ ∈ (↑s : Set α)
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₅ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α) (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)),
  False
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
refine₁ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
    s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
refine₂ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
refine₃ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
refine₄ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∨ aᶜ ∈ (↑s : Set α)
refine₅ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α)
    (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)), False
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₆ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α) (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)),
  (↑(insert a s) : Set α).Intersecting
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm
-- Lean 4 proof step from Mathlib.Combinatorics.SetFamily.Intersecting (goal)
Goal:
α : Type u_1
inst : BooleanAlgebra α
inst_1 : Nontrivial α
inst_2 : Fintype α
s : Finset α
hs : (↑s : Set α).Intersecting
refine : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), 2 * #s = Fintype.card α
refine₁ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t),
    s.disjUnion (map { toFun := compl, inj' := ⋯ : α ↪ α } s) ⋯ = Finset.univ
refine₂ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t), (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α) = univ
refine₃ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∪ compl ⁻¹' (↑s : Set α)
refine₄ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α), a ∈ (↑s : Set α) ∨ aᶜ ∈ (↑s : Set α)
refine₅ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α)
    (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)), False
refine₆ :
  ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α)
    (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)), (↑(insert a s) : Set α).Intersecting
⊢ (∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) ↔ 2 * #s = Fintype.card α
have refine₇ : ∀ (h : ∀ (t : Finset α), (↑t : Set α).Intersecting → s ⊆ t → s = t) (a : α) (ha : a ∉ (↑s : Set α) ∧ aᶜ ∉ (↑s : Set α)),
  (insert a (↑s : Set α)).Intersecting
Tactic: classical
    refine ⟨fun h ↦ ?_, fun h t ht hst ↦ Finset.eq_of_subset_of_card_le hst <|
      Nat.le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) Nat.two_pos⟩
    suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by
      rw [Fintype.card, ← this, Nat.two_mul, card_disjUnion, card_map]
    rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,
      image_eq_preimage_of_inverse compl_compl compl_compl]
    refine eq_univ_of_forall fun a => ?_
    simp_rw [mem_union, mem_preimage]
    by_contra! ha
    refine s.ne_insert_of_notMem _ ha.1 (h _ ?_ <| s.subset_insert _)
    rw [coe_insert]
    refine hs.insert ?_ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb
    rintro rfl
    have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)
    rw [compl_bot] at ha
    rw [coe_eq_empty.1 ((hs.isUpperSet' h).top_notMem.1 ha.2)] at this
    exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm