What do you do if your sister is a chronic liar and self-centered?

I am an adult now, but my oldest sister is a chronic liar when I was growing up. Her lying has isolated many family members. She is also self-centered to the point that she will only do what benefits her. Recently my middle sister died and I am close to her. My oldest sister did not attend the arrangements for my sister because she had a convention to go to for her business. I said to her, "can't your husband go and fill in for you and you go up the next day". She responded that he would just mess it up. Needless to say we carried on without her. This was the last straw for me and I do not have anything to do with her at present.. I am so distraught over this situation and am crying all the time since I cut my oldest sister out of my life. But what can I do since she won't change and only means you some kind of harm?