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}\bigtriangleup Y_{s}|=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.