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 unbounded Separation, and to prove "there exist infinitely many pairwise nonequinumerous infinite sets" requires Replacement. 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