How do I most straightforwardly prove that if an ultrafilter on a cardinal $\lambda$ is principal, then it is of the form all subsets of $\lambda$ containing $\alpha$ for some $\alpha<\lambda$?
What is the most direct reason why such an ultrafilter cannot contain all supersets of a non-singleton set.
I'm assuming your definition of principal is $\{F\subset E\mid A\subset F\}$ for some $A$ ? (it is often in the definition of "principal" that $A=\{x\}$)
In that case, since $\mathcal{U}$ is an ultrafilter, if $B\subset A$, $B\neq A,\emptyset$, what do you make of the question "does $B\in \mathcal U$ ?"