The meaning of E[X|Y] = E[X]

In class today it was mentioned that when E(X|Y) = E(X) , the two random variables X and Y are uncorrelated, i.e. cov(X, Y) = 0.

I’m curious as to how this property works.

Why are the two variables X and Y uncorrelated when E(X|Y) = E(X)? The professor just mentioned it as passing as if it was a natural thing, but I can’t seem to understand how this property is proved.

Could anyone tell me why E(X|Y) = E(X) means cov(X,Y) = 0 ? Thanks.