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 .