My Calculus of Probability is very rusty, so I need your advice.

I have two independent uniformly distributed random variables $X$ and $Y$ with densities:
$$\begin{split} f\_X(\omega) &:= \begin{cases} \frac{1}{\Phi} &\text{, if } \xi\leq \omega < \xi + \Phi \\ 0 &\text{, otherwise}\end{cases} \\
f\_Y (\omega) &:= \begin{cases} \frac{1}{\Psi} &\text{, if } \eta \leq \omega < \eta + \Psi \\ 0 &\text{, otherwise}\end{cases} 
\end{split}$$
(with $\xi,\eta \in \mathbb{R}$ and $\Phi,\Psi >0$) and I'm interested in finding sufficient conditions for $\mathbb{P}(X<Y) >0$.

Now, since $\mathbb{P}(X<Y) = \mathbb{P}(X-Y<0)$ and since $X-Y$ is a r.v. with density $f\_{X-Y} = f\_X \* f\_{-Y}$ (where $\*$ is the convolution operator), I find that a sufficient condition to $\mathbb{P}(X<Y) >0$ is that $0$ lies to the right of the minimum of the support of $f\_{X-Y}$, for in this case:
$$\mathbb{P}(X<Y) = \mathbb{P}(X-Y<0) = \int\_{-\infty}^0 f\_{X-Y}(\omega)\ \text{d} \omega > 0\; .$$
Is that correct?

Moreover, I find that $f\_{X-Y}$ is a trapezoidal density function. Could someone check if:
$$\begin{split} f\_{X-Y} (\omega) &= \begin{cases} 0 &\text{, if } \omega < \xi - \eta - \Psi\\
\frac{\omega + \eta + \Psi -\xi}{\Phi \Psi} &\text{, if } \xi - \eta - \Psi \leq \omega < \xi + \Phi - \eta - \Psi\\
\frac{1}{\Psi} &\text{, if } \xi + \Phi - \eta - \Psi \leq \omega < \xi - \eta\\
\frac{\xi + \Phi - \eta -\Psi - \omega}{\Phi \Psi} &\text{, if } \xi - \eta \leq \omega < \xi + \Phi - \eta\\
0 &\text{, if } \xi +\Phi -\eta \leq \omega
\end{cases}\qquad \text{ if } \Psi \geq \Phi \\
f\_{X-Y} (\omega) &= \begin{cases} 0 &\text{, if } \omega < \xi - \eta - \Psi\\
\frac{\omega + \eta + \Psi -\xi}{\Phi \Psi} &\text{, if } \xi - \eta - \Psi \leq \omega < \xi - \eta\\
\frac{1}{\Phi} &\text{, if } \xi - \eta \leq \omega < \xi + \Phi - \eta - \Psi\\
\frac{\xi + \Phi - \eta -\Psi - \omega}{\Phi \Psi} &\text{, if } \xi + \Phi - \eta - \Psi \leq \omega < \xi + \Phi - \eta\\
0 &\text{, if } \xi +\Phi -\eta \leq \omega
\end{cases}\qquad \text{ when } \Phi > \Psi
\end{split}$$
is correct?
These computations imply that $\mathbb{P}(X<Y)>0$ iff $\xi - \eta -\Psi < 0$, i.e. iff $\xi < \eta +\Psi$, don't they?

Is there a simpler way to solve the problem?

Thanks a lot in advance.