English
If a single component s i is Infinite, then the union over i of s i is Infinite.
Русский
Если хотя бы одно из множеств s i бесконечно, то объединение ⋃ i s i бесконечно.
LaTeX
$$$\\operatorname{Infinite}\\left(\\bigcup_i s_i\\right)\\;\\text{given } (s_i)\\text{ Infinite for some } i$$$
Lean4
/-- If sets `s i` are finite for all `i` from a finite set `t` and are empty for `i ∉ t`, then the
union `⋃ i, s i` is a finite set. -/
theorem iUnion {ι : Type*} {s : ι → Set α} {t : Set ι} (ht : t.Finite) (hs : ∀ i ∈ t, (s i).Finite)
(he : ∀ i, i ∉ t → s i = ∅) : (⋃ i, s i).Finite :=
by
suffices ⋃ i, s i ⊆ ⋃ i ∈ t, s i by exact (ht.biUnion hs).subset this
refine iUnion_subset fun i x hx => ?_
by_cases hi : i ∈ t
· exact mem_biUnion hi hx
· rw [he i hi, mem_empty_iff_false] at hx
contradiction
