This very matter is  discussed in  depth by Mathias,  in Chapter 9 of his  **The Stength of Mac Lane  Set Theory** https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows  that to prove "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets" requires some use  of  unbounded Separation, and to prove "there exist infinitely many pairwise nonequinumerous infinite sets" requires some use  of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy.


Meanwhile, some of the motivation of this question resonates with mine in  posing these questions:

http://mathoverflow.net/questions/120598/when-must-it-be-sets-rather-than-proper-classes-or-vice-versa-outside-of-fou

http://mathoverflow.net/questions/117910/can-one-exhibit-an-explicit-kuratowski-infinite-set-without-invoking-replacement