I'm trying to prove the equivalence of the following two (weaker) definitions of finiteness. I know that these definitions are equivalent to the standard definition of finiteness if the axiom of choice holds, but I'm trying to prove they are equivalent to one another, possibly without the axiom of choice. I'm not sure if it's possible though.
1) A set $S$ is finite if there is no bijection between $S$ and any proper subset of $S$.
2) A set $S$ is finite if there exists an injection from $S$ to $\mathbb{N}$ but no injection from $\mathbb{N}$ to $S$.