I'd wait another year or two before pruning, and even then, only prune them off if they are a nuisance with being in your way. No, removing the lower branches of a tree does not encourage more upward growth, it just makes for a weaker, spindlier tree with less ability to grow a good strong trunk.

Resin

I agree with Resin on the pruning, but should you decide that you need to remove an offending small branch or two, please call your local extension office to find out the appropriate time of year in your location. Bark beetles and borers are very attracted to newly pruned trees, and you need to find out when it is safe to prune.

Prune elms during winter to avoid attracting the beetles, which can carry DED. True, Princeton has resistance to this disease, but precautions still are warranted.

+oM