eqv_classes_of_disjoint_union — Mathlib

English

A variant of the previous statement with an alternative proof path.

Русский

Вариант предыдущего утверждения с альтернативным ходом доказательства.

LaTeX

$$A second form of the disjoint-union lemma: exists uniqueness via H and h-elimination.$$

Lean4

/-- A set of disjoint sets covering α partition α (classical). -/
theorem eqv_classes_of_disjoint_union {c : Set (Set α)} (hu : Set.sUnion c = @Set.univ α) (H : c.PairwiseDisjoint id)
    (a) : ∃! b ∈ c, a ∈ b :=
  let ⟨b, hc, ha⟩ := Set.mem_sUnion.1 <| show a ∈ _ by rw [hu]; exact Set.mem_univ a
  ExistsUnique.intro b ⟨hc, ha⟩ fun _ hc' => H.elim_set hc'.1 hc _ hc'.2 ha

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