Women are Sexist Too

A lot of people are under the false impression that only men are sexist. That statement is sexist. Recently I've seen a lot of sexism coming from women, and just like sexism coming from men, it needs to stop.
I was talking with a girl from school today. We were talking about what we want to do when we graduate. When I told her I wanted to be a doctor she immediately corrected me. She asked "you mean a nurse right?" Then went on to say "Doctors are men."
This is just one example of sexism coming from women. Why can't a doctor be a woman? I know lots of female doctors, and male nurses. But for some reason, when you hear 'doctor' you think man, and when you hear 'nurse' you think woman.
Another example of sexism from women was pointed out in a Facebook post by an old friend of mine. Her point was about how adults act around children and babies.
She pointed out that it is acceptable for women to coo over babies, in fact, they are expected too. Men, on the other hand, are not "supposed" to do this. In fact, when they do play with children and babies, its makes them seem out of the ordinary and 'special'. Why? Because of how our society works. Because sexism runs so deep in our culture that its hard to imagine a man taking care of his child and a woman being the breadwinner of the family.
Women accept this as a truth, and that's why women are sexist. They may not even realize it.
What do you think? Do you think that women are being sexist? Or do you think that this is just how life is? Do you think that we will ever see a change?