You don't engage with them any further than necessary. It's clear if they don't want to listen to the rules that you aren't going to be able to persuade them. If you're not a person in a position for power or authority (and if you are you can just remove them) then they're probably not going to listen to you without causing a huge fuss or even argument, and even then that might not go in your favour.
Go and get someone who should be dealing with rule breaking, an officer, conductor or anyone else who has at least some authority and let them deal with it. They probably even have training on how to deal with this type of person, and you could observe and learn from what they do.
I personally feel as though there is no point trying to save face in front of that sort of person, they aren't worth your time or effort. If you've explained what they're doing is wrong and they still do it just leave them to, go and get someone more official, and let them deal with them and take the relevant action. Just as a personal preference I would even reply if they refuse, I would just go and seek the relevant authority/personal.