Yes. You started with -0=0, which is what you're trying to prove in the first place, and you end up with the true statement 0=0. This isn't a valid form of proof unless you can show that each step is reversible, so we have to start from the bottom and see if that's true.

0=0. So far so good...

-0+0=0+0. This is true because -0=0, but unfortunately that's what you're trying to prove, so you can't use this step.

I still say that any proof over one line is unnecessarily long. 0+0=0 is all you need. Think about what it says. If you add 0 to itself you get 0, and isn't 0 the additive identity in a ring?