Actually, in the US, the "who was first?" date is the date of invention, not the date of patent application submission. Of course, it's not always easy to prove that you invented something on a particular date, say, a date before your competitor submitted their patent application. It's especially difficult if you've been developing an idea in secret for some time before applying for the patent (though I believe there's an 18 month grace period before the USPTO will publish your patent application filing).
Or so a patent lawyer told me, anyway. It's possible I misunderstood something.
The accepted method is to log it into a patent notebook (bound book with numbered pages), and get it co-signed by someone knowledgeable enough to testify in court about the content. Signature should say 'read and understood'.
This book makes good reading:
Laser: The Inventor, the Nobel Laureate, and the Thirty-Year Patent War
A much better way to establish priority is to file a provisional patent, good for a year.
The book "Patent it yourself" makes good reading - not as good as working with a patent attorney, but a lot cheaper.
Out of curiosity: could you prove it by sending an email to a system you don't control, like Gmail? If you kept gmail logs of your creations / thoughts, and can't fake the date, it seems to me that'd be proof.
Or can you fake the date with Gmail? If so, know of a possible alternative? I'll probably end up using it.