How would we prove that infinite sets have at least a cardinality of aleph naught?
I am not sure to which extent this fulfills OPs needs, but perhaps it is useful to have this somewhere for reference. (I guess this can be found, in some form, in several other questions at this site.)
Several definitions of finite and infinite sets are used in mathematics. The following result, taken from H. Herrlich: Axiom of Choice, p.44, shows that one of them, called Dedekind-infinite, is equivalent to having cardinality at least $\aleph_0$. You can find a detailed proof there.