Can conditional expectation be used to prove the following result?

Question

Let $X$ and $Y$ be two random variables. Is it possible to use the law of total expectation to say

$$ \mathbb{E}[X - \mathbb{E}[X \mid Y]] = \mathbb{E}[\mathbb{E}[X - \mathbb{E}[X \mid Y] \mid Y] = 0. $$

It seems right but I believe I might be committing some crime.

Thanks

Answer 1

Note that $$\mathbb{E}[X-\mathbb{E}[X \mid Y]]=\mathbb{E}[X]-\mathbb{E}[\mathbb{E}[X\mid Y]]=\mathbb{E}[X]-\mathbb{E}[X]=0$$ by linearity and the law of total expectation.

Of course, since the above is zero, any expectation (conditional or unconditional) of the above is also zero.