A set is infinite iff there is a one-to-one correspondent with one of its proper subsets? Maxwell Rosenlicht claims in "Introduction to analysis" that a set is infinite if and only if it may be placed into one-to-one correspondence with a proper subset of itself.
He says this is self-evident because a finite set cannot be placed into a one-to-one correspondence with a proper subset of itself (because it has fewer elements), and whilst this is reasonable - I cannot follow Rosenlicht in that "the above therefore follows obviously". Why must a set be infinite just because of some property of finite sets?

If set $S$ is finite then there is no such proper subset. This means that if there is such proper subset for $S$ then $S$ is infinite. It's just $(A \implies B) \iff (\neg B \implies \neg A)$ for $A = [\text{$S$ is finite}]$ and $B = [\text{no such proper subset exists}]$.
