It proves itself if you define "I" as the same as the experiencer of the thinking, but "I" (as many other complex word) is much more overloaded.

Unfortunately all our word definitions seems shaky, if we want to describe something that is the base requirement of those very definitions.

Maybe the best we can do is to deconstruct the above using more simple or base terms, but the meaning of those terms maybe also depends on the content of experience not the mere fact of experience:

I experience thought -> experience of thoughts exists -> experience exists -> something exists

So upon experiencing thought you may conclude that "something exists", or "there IS something"...