Girls are going to rule the world

So I learned in SexEd that the male chromosome, y, is getting smaller and smaller until it basically vanishes. The female chromosome, X, is larger than the y chromosome, and not getting any smaller.

With modern technology, like artificial insemination, girls just need to collect as much sperm as they want, and keep spawning more humans to keep the population alive in the future.

I know I won't really live to see the day, but how much would change culturally?
You think girls are going to be the all-power imposing the patriarch has been on society?