Need advice: static imploding because of cheater.

[Rilifane]

Do I really need to write it into rules "don't cheat"? Do I also have to have a rule saying "don't be racist" or "don't try to hit on the women in the static"?

You have to be clear about having a zero-tolerance policy and kicking the people immediately without saying a word to everyone else, yes. You do. That is part of being a static leader: being transparent instead of just stomping your foot, crossing your arms and doing what you want.
You handed it terribly and what happens now just further proves it.