If $E(Y)=0$, $Cov(Y, E(Y|X))\geq0$ for any random variable $X$

Let $Y$ be a random variable with mean zero and finite variance, and let $X$ be any other random variable. Using the fact that $\mathbb{E}[Y]=0$ and the law of iterated expectation, we have $$\text{Cov}(Y, \mathbb{E}[Y|X])=\mathbb{E}[(Y-\mathbb{E}[Y])(\mathbb{E}[Y|X]-\mathbb{E}[\mathbb{E}[Y|X]])]=\mathbb{E}[Y\mathbb{E}[Y|X]].$$ I am tempted to say this is the expectation of $Y^2$, which would settle the question, but I don't know how to justify it (or even if it's true in the first place).