eq_pure_of_finite_mem — Mathlib

English

Let f be an ultrafilter on α and s ⊆ α finite with s ∈ f. Then there exists x ∈ s such that f = pure x.

Русский

Пусть f — ультрафильтр на α, а s ⊆ α конечно и s ∈ f. Тогда существует x ∈ s such that f = pure x.

LaTeX

$$Let α be a type, f a Ultrafilter α, s ⊆ α with s.Finite and s ∈ f. Then ∃ x ∈ s, f = pure x.$$

Lean4

theorem eq_pure_of_finite_mem (h : s.Finite) (h' : s ∈ f) : ∃ x ∈ s, f = pure x :=
  by
  rw [← biUnion_of_singleton s] at h'
  rcases (Ultrafilter.finite_biUnion_mem_iff h).mp h' with ⟨a, has, haf⟩
  exact ⟨a, has, eq_of_le (Filter.le_pure_iff.2 haf)⟩

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