Given an

element `x`∈X, construct the

set of all sets containing

`x`:

E = E_{x} = {A⊆X | `x`∈A}

E is known as a

*principal ultrafilter*.

Then it is easy to see that E is a filter (see cjeris' writeup!).

And if F!=E were a filter containing E, then F would contain some set B not containing `x` and also {`x`}, so it would contain their intersection -- the empty set. That's a contradiction, so E is maximal, hence an ultrafilter.

Principal ultrafilters are the only constructible ultrafilters.