# Inequality for joint probabilities of dependent random vectors

Let $$X$$ and $$Y$$ be two dependent random vectors in $$\mathbb{R}^d$$, such that $$X\neq Y$$ with probability 1, whose joint probability measure has density $$\mu(x,y)$$ with respect to the Lebesgue measure. For measurable sets $$A$$ and $$B$$, does the inequality $$\mathbb{P}(X-Y \in A, Y\in B) \leq \sup _{t\in B}\mathbb{P}(X \in A+t)$$ hold true? Herein, operations are meant elementwise and $$A+t:=\{x+t:x \in A\}$$.

I was thinking to go through something like:

$$\mathbb{P}(X-Y \in A, Y\in B)= \int_{B}\int_{A+y}\mu(x,y)dxdy\\ \leq \sup_{t\in B}\int_{B}\int_{A+t}\mu(x,y)dxdy\\ =\sup_{t\in B}\int_{A+t}\int_B\mu(x,y)dydx\\ \leq \sup _{t\in B}\int_{A+t}\mu(x)dx\\ =\sup _{t\in B}\mathbb{P}(X \in A+t)$$

where $$\mu(x)$$ denote the marginal density of $$X$$, but I have doubts about the second and third lines, I'm not sure they're correct. I pass from the third to the fourth line by using the fact that $$\mu(x,y)$$ is nonnegative and $$\int_B\mu(x,y)dy \leq \int_{\mathbb{R}^d}\mu(x,y)dy=\mu(x)$$.