"Sexism towards men doesn't exist"
Ask any man who's gotten divorced and he'll tell you you're full of shit.
But it doesn't.
And you can thank male dominated society for any ill situations towards men. Men have put out this odd belief that only women are capable of caring for a child or they're incapable of supporting themselves. So when a divorce occurs and the woman is given custody or a large sum of money, blame men's irrational belief that men are the sole "bread winners" and aren't as nurturing as parents.
If there was a burning building with 3 people (1 woman and 2 men) trapped in there and only once could be saved, the woman will always be the first priority. Why is that? There is no valid or rational reason. You're silly for not thinking sexism towards men exist. It most certainly does. I've experienced it first hand plenty of times.
Woman would be saved first solely on the basis that they are seen as incapable, weak, and irrational. You can thank men for that belief. The woman would be saved especially if she is a mother because society has the belief that women are the true parents and belong with raising children.
Sexism towards men does not exist. PREJUDICE does however. Sexism and prejudice are very different. Sexism (same with racism) is a systematic oppression by the dominant culture against the subordinate culture. (i.e. whites against minorities, men against women)