In fact, if time began with the big bang (and presupposing that everything started with the big bang), then everything is eternal (or at least everything has always existed up to the present). Your contradiction is in simultaneously assuming both that there was and that there wasn't a "time before time". If time began with the big bang, then there can be no "before" the big bang.
Congratulations! So far, you are the only one brave enough to think about the problem. You are correct in your observation that such a statement as the cosmologists make, "Before the Big Bang there was no time," is logically contradictory since the word "before" requires the existence of time. This, however, is only a problem for materialists who believe that matter and its motion are all there is. It is not a problem for Creationists who believe that there is a reality beyond this physical universe.
If, therefore, there is a reality outside of this universe there can be a cosmological "before time."
Good question, good thinking.