The alternative set theory of Vopenka also has only these two cardinals. The alternative set theory also exploits this. Notice that we get a unique empty class in addition to any non-classes. I is a Dedekind infinite set. It is straightforward to prove that I has any concrete finite number of distinct elements.
All proper classes are the same size.
The empty class is a set. There is a set with one element. Then every proper class would have one element. Note that we have actually proved that every singleton is a set. As soon as 2 is shown to be a set, we have pairing, and the ability to represent relations as sets of ordered pairs and quantify over relations.