You can get that every integral domain is a field with fewer words by using a higher powered set theory result -- injections on finite sets are also surjections. The cancellation property says multiplication by any element is an injection, so it is also a surjection, i.e., 1 is in the range, so that gives you the multiplicative inverse.
The correct statement is that an injection from a finite set to itself is a surjection. The converse is true, too. A surjection from a finite set to itself is an injection.
In this case, multiplication by any nonzero fixed element of the ring is an injection from the ring to itself. Any injection from a finite set to itself is indeed a surjection (and so also a bijection).
