In set theory, the **Cantor-Bernstein-Schroeder theorem** is the theorem that for if there exist injective functions `f` : `A` → `B` and `g` : `B` → `A` between the sets `A` and `B`, then there exists a bijective function `h` : `A` → `B`.
In effect, this means that if the cardinality of *A* is less than or equal to that of *B*, and the cardinality of *B* is less than or equal to that of *A*, then *A* and *B* have the same cardinality. This is obviously a very desirable feature of the ordering of cardinal numbers.

Here is a proof [due to Eilenberg?]:

Let

- ,

One can then check that *h* : *A* → *B* is the desired bijection.

An earlier proof by Cantor relied, in effect, on the axiom of choice by inferring the result as a corollary of the well-ordering theorem. The argument given above shows that the result can be proved without the axiom of choice.