I am trying to understand the Random Priority version of the parallel Luby's algorithm to find a maximal independent set (MIS). In the analysis steps here, the first bullet claims that a node v removes it's neighbor w if r(v) < r(w) and r(v) < r(x), where x is a neighbor of w and r(x) is the random number of node x.
I do not understand why r(v) < r(x) needs to be true for v to remove it's neighbor w. Isn't the fact that r(v) is less than that of all it's neighbors enough for it to be included in the MIS?