I don't think this is true. Are we assuming that every universe has an infinite number of civilizations? It's impossible to have uniform probability across an infinite set of discrete probabilities, because every outcome would have a probability of 0.

So with no other knowledge, it's slightly more likely that we are the first civilization than the 5 millionth.

Of course it's tricky to talk about the "first" civilization in the face of relativistic time, but I'm assuming that there is some way to have an ordering of all civilizations.

Assuming there are very many civilizations in our universe, the odds that we're the first are tremendously lower than the odds that we're not. I think that's the point.