Should i cut all contact with my friend?

(The names I use in my story are not, the people's real names)

Right now I'm in year 10, I've had this friend, Emily, since year 2. We were in the same class until year 7. We both changes schools after that. But we still see each other often.
For as long as I can remember, she has always put me down and spoken rudely to me - she has always made me feel small. We were out together the other day and I told her that I was meeting up with our mutual friend, Tiana, that I have not seen for two years (neither has she). She invited herself and made plans for us, I told her politely that I would rather meet up with her alone because that way it would be easier to catch up (the real reason was; that I don't want Tiana and my relationship to be the same as me and Emily's). She got annoyed and threw a tantrum (and that was not the first time, she does that all the time). I told her that I was not in the mood for her tantrum and that I would rather go home. Here she got very annoyed and now we are in a big fight.
But this fight has made me think, maybe our relationship just isn't very healthy and maybe I should cut contact with her. I have told her how I feel before but nothing has happened. Maybe this fight sounds stupid, but it is just one out of many.

So my question is: Should I cut contact with Emily?