Theorems · Theorem · order theory

Set.PairwiseDisjoint.finite_biUnion_iff

∀ {α : Type u} {β : Type v} {f : β → Set α} {s : Set β},
  s.PairwiseDisjoint f → ((⋃ i ∈ s, f i).Finite ↔ (∀ i ∈ s, (f i).Finite) ∧ {i | i ∈ s ∧ (f i).Nonempty}.Finite)

An indexed union of pairwise disjoint sets is finite iff all sets are finite, and all but finitely many are empty.

Defined in

Mathlib.Data.Set.Finite.Lattice

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound