The law of iterated expectations

Problem

If $X$ and $Y$ are two random vectors then $E\left( Y \mid X \right)$ is a new random vector of the same dimension as $Y$ . The law of iterated expectations states that if you take the expectation of the random vector $E\left( Y \mid X \right)$ you get the unconditional expectation $E\left( Y \right)$ ,

$$E\left( E\left( Y \mid X \right) \right)=E\left( Y \right)$$

Show that

$$E\left( Y \mid X \right)=0 \implies E\left( Y \right)=0$$

Note that the opposite is not true.

Solution

$$E\left( Y \right)=E\left( E\left( Y \mid X \right) \right)=E\left( 0 \right)=0$$