Equivalent Conditions for Dedekind-Infinite Set

Theorem
For a set $S$, the following conditions are equivalent:
 * $\left({1}\right):\quad$ $S$ is Dedekind-infinite.
 * $\left({2}\right):\quad$ $S$ has a countably infinite subset.

The above equivalence can be proven in Zermelo-Fraenkel set theory.

If the axiom of countable choice is accepted, then it can be proven that the following condition is also equivalent to the above two:
 * $\left({3}\right):\quad$ $S$ is infinite.