Is it normal to do things purely for my ego?

I've been single for a long time because I don't really care about having friends or forming a relationship, but the fact that I don't care bothers me because most people do, and I like to think I'm at least as good as most people. So I want to get and hold a relationship basically just to prove to myself that I can, and not because I think being in a relationship will make me happier.

I realized I have a job for exactly the same reason. I don't really care that much about it, it's not a dream job, but I wanted to prove to myself that I could land and work a "real" job so I worked through a degree and here I am.

It makes me wonder, maybe everything I do is for my ego, and I just didn't think about it that way? Is this normal, or am I just egocentric?