Hmmm. I see your point. And I have no problem being wrong. And I also admit to thinking about things unusually.
But my question still makes sense as an answer - to me anyway. A carrot doesn't think (as far as we know), but I don't know how a carrot can be proven to exist, even if, by using that statement, it can be 'proven' that we exist. If the above statement is true, that because we think we exist, than how can we prove anything else exists? I think therefore a carrot exists? I think therefore angels exist? (There are many people who claim to have seen them, which is why I'm using that as an example)
Also, don't we believe we exist because that's the only option we are aware of?
I'm not necessarily saying that I believe we don't exist, but I am saying that the above statement is an opinion, because we just believe we exist because that's been our only way to rationalize our existence.