Define pretty sets in LaTeX; esp., how to do the condition-separator?

I sometimes do something a little more evil

\documentclass[a4paper]{memoir}
\usepackage{mathtools}
\DeclarePairedDelimiterX\Set[2]{\lbrace}{\rbrace}%
 { #1 \,\delimsize| \,\mathopen{} #2 }
\begin{document}
\[
\Set*{ x }{ x>0 }
\]
\end{document}

and thus hide the symbol inside the construction. Both a starred and non-starred \Set is created with \Set* autoscaling braces and vertical line is used.

Update August 2014: I no longer recommend a two argument solution, but rather this as I feel it gives an interface much closer to the mathematical meaning

\documentclass[a4paper]{memoir}
\usepackage{mathtools}
\providecommand\given{} % so it exists
\newcommand\SetSymbol[1][]{
   \nonscript\,#1\vert \allowbreak \nonscript\,\mathopen{}}
\DeclarePairedDelimiterX\Set[1]{\lbrace}{\rbrace}%
 { \renewcommand\given{\SetSymbol[\delimsize]} #1 }
\begin{document}
\[
\Set*{ x \given x>0 }
\]
\end{document}

BTW: notice the added \mathopen{} withput it \Set{ X \given -a < x < a} would give us the wrong -a (a subtraction minus, not a sign minus)

Update March 2015: Moved the \allowbreak in front of the proceeding inserted space. Then that space disappears if a line break happens.

You should use \mid instead of |. And use \colon instead of : after the existential quantifier (although I would just put a thinspace).

The braket package contains a \set command which does what you want :

\set{\alpha \in \Sigma^\omega | \exists^\omega n : \alpha [0,n] \notin \Sigma^* a_1 \Sigma^* a_2 \dotsb a_n \Sigma^*}

It also defines \Set, where the first | is expandable, which allows to properly typeset sets like

\Set{x| x\in\mathbb R, |x|<\frac12 }