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.