There are a few additional remarks on the Schröder-Bernstein theorem worth mentioning. I will expand on some of them later, in the context of descriptive set theory.

The** dual Schröder-Bernstein theorem **(**dual S-B**)** **is the statement** ***“Whenever are sets and there are surjections from onto and from onto then there is a bijection between and .”*

* This follows from the axiom of choice. In fact, is equivalent to: Any surjective function admits a right inverse. So the dual S-B follows from choice and the S-B theorem.

* The proofs of S-B actually show that if one has injections and , then one has a bijection contained in . So the argument above gives the same strengthened version of the dual S-B. Actually, over , this strengthened version *implies* choice. This is in Bernhard Banaschewski, Gregory H. Moore, *The dual Cantor-Bernstein theorem and the partition principle*, Notre Dame J. Formal Logic **31 (3)**, (1990), 375-381.

* If is onto, then there is 1-1, so if there are surjections in both directions between and , then and have the same size. Of course, this is possible even if and do not.

**Open question.** **(****)** *Does the dual Schröder-Bernstein theorem imply the axiom of choice?*

* The dual S-B is not a theorem of .