Alternatively, one can use the (Cantor-)Schröder-Bernstein property of sets:
Let A, B be sets.
1. Write A ≤ B if there is an injective map from A to B. Then A ≤ B ≤ A gives A = B.
2. Write A ≤ B if there is an surjective map from B to A. Then A ≤ B ≤ A gives A = B.
Actually, the CSB property is only part 1. Part 2 is dependent on the axiom of choice, I believe.
Both are CSB properties. The first one is the well-known CSB theorem, the latter is its dual (and indeed requires the axiom of choice). In general, a CSB property in its most general form (that is known to me) is a statement of the following type:
Let C be some category. For objects A, B write A ≤ B is there is a monomorphism from A to B. We say that C satisfies the CSB property if A ≤ B ≤ A implies that A, B are isomorphic in C.
One could replace "monomorphism" by any property of morphisms and "isomorphic" by "equivalent in regard to some equivalence relation on objects". But I never saw those generalisations in actual use.
If C are all sets, then "monomorphism" simply means "injective" and we get the first one from above.
Now let C be the dual category of sets. Then a monomorphism in C is an epimorphism of sets, i.e. a surjective map in the opposite direction. This gives the second one.
Wikipedia only really talks about injective versions in
http://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_property, but their formulation is actually just what I wrote above.
