You want your intent to be as direct as possible. You are leaving, plain and simple. There's no need to mention where you're going. You can discuss that with your boss (though you're not obligated to do so) and your colleagues if you want, but there's little purpose for including that in your resignation letter. That's especially true if you're moving over to a competitor__you don't want to create ill will in this letter or cause any drama.
Or meet up with said colleagues at an industry event, because the world is awkwardly small. So it's in your best interest to make sure that your resignation letter is clear, concise, and polite. Here are some best practices you can follow to make your big exit a professional one.