The Cantor–Schröder–Bernstein Theorem for $$\infty $$-groupoids

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