Social choice theory
You are correct that it is not possible to violate Weak Pareto and Nondictatorship at the same time.  But your explanation (second paragraph) is a bit muddled.  Here is how I would put it.

To prove, "it is not possible to violate Weak Pareto and Nondictatorship at the same time", it suffices to prove:  "Any Dictatorship satisfies Weak Pareto."  To prove this, suppose F is a social choice rule in which voter i is a dictator.  So for any alternatives a and b, if i prefers a over b, then so does the output of rule F.  To see that F satisfies weak Pareto, let a and b be social alternatives, and suppose that everyone prefers a over b.  Then in particular, i does.  Thus, the output of F prefers a over b (because i is a dictator in F).  Therefore, F satisfies Weak Pareto.