Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.

Is anyone familiar and can refer me to a result(or a counterexample) which states that

${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}(\omega)\bigtriangleup Y_{s}(\omega)|=0 $ for all $t\in \mathbb{R}$ a.s?

A different yet equivalent formulation of this is

$\bigtriangleup X_{t}=0$ or $\bigtriangleup Y_{t}=0$ a.s. for all $t\in \mathbb{R}$

In words, every two independent Levy processes have no simultaneous jumps a.s. I know it holds for independent Poisson processes and I'm wondering if it generalizes.