Given that $E[X_1]≥E[X_2]$, can we say $P[X_2≤C]≥P[X_1≤C] $?

Question

Let $X_1$ and $X_2$ be two random variables defined over the same domain $\mathcal{D}$. Let $C \in \mathcal{D}$ be a constant.

Given that $E[X_1] \geq E[X_2]$, then does the following assertion hold for any particular type of probability distribution over $X_1$ and $X_2$ or any particular subset of $C$: $$P[X_2\leq C] \geq P[X_1 \leq C]$$

where E stands for expectation and P for probability.

Thanks!