Consider a finite time interval, let $J_i$ denote the set of jumps of the $i$'th process, and let $J_{i,n}$ denote the set of jumps of the $i$'th process of size $(\frac 1 n,\frac 1{n-1}]$. Note that $J_{i,n}$ is finite a.s. and therefore $J_{i,n}\cap J_{j,m}=\emptyset$ a.s. for all $n,m$ and $i\neq j$. Since there are countably many such intersections, it follows that $J_i\cap J_j = \emptyset$ a.s.

Consider a finite time interval, let $J_i$ denote the set of (times of) jumps of the $i$'th process, and let $J_{i,n}$ denote the set of jumps of the $i$'th process of size $(\frac 1 n,\frac 1{n-1}]$. Note that $J_{i,n}$ is finite a.s. and therefore $J_{i,n}\cap J_{j,m}=\emptyset$ a.s. for all $n,m$ and $i\neq j$. Since there are countably many such intersections, it follows that $J_i\cap J_j = \emptyset$ a.s.

Consider a finite interval, let $J_i$ denote the set of jumps of the $i$'th process, and let $J_{i,n}$ denote the set of jumps of the $i$'th process of size $(\frac 1 n,\frac 1{n-1}]$. Note that $J_{i,n}$ is finite a.s. and therefore $J_{i,n}\cap J_{j,m}=\emptyset$ a.s. for all $n,m$ and $i\neq j$. Since there are countably many such intersections, it follows that $J_i\cap J_j = \emptyset$ a.s.

Consider a finite time interval, let $J_i$ denote the set of jumps of the $i$'th process, and let $J_{i,n}$ denote the set of jumps of the $i$'th process of size $(\frac 1 n,\frac 1{n-1}]$. Note that $J_{i,n}$ is finite a.s. and therefore $J_{i,n}\cap J_{j,m}=\emptyset$ a.s. for all $n,m$ and $i\neq j$. Since there are countably many such intersections, it follows that $J_i\cap J_j = \emptyset$ a.s.