is this for a set theory class? i just want to know how rigorous your proofs have to be. here are some simple approaches, they probably wouldn't hold up in a set theory class, but they'd be ok in most other classes:

(1) let x be fixed and do induction on y

(2) a proof by contradiction. show that is false if y is not zero.

(3) since y is a natural number that is not zero, it must be greater than 0 (since zero is the least element). so we have y > 0. now add x to both sides.