The Cantor–Schröder–Bernstein Theorem for ∞-groupoids

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or ∞ -groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean ∞ -topos.