 Cantor-Schröder-Bernstein theorem

If there are injections $X\to Y$ and $Y\to X$, then there is a bijection $X\cong Y$.

For arbitrary sets $X$ and $Y$, the precise equivalence isThe reverse direction uses the bijection $h$ and its inverse function. Neither set is assumed to be a subset of the other: the hypothesis concerns functions, rather than containments. The forward direction holds in ZF without the axiom of choice.^[ref]

This makes cardinality comparison an antisymmetric relation: if each set is no larger than the other in the sense of an injection, they have equal cardinality. It also gives a practical strategy: build two convenient injections and let the theorem supply the bijection. The Injection-chain proof of the Cantor-Schröder-Bernstein theorem explains how to combine them without losing an output or pairing two inputs with the same output.