Set builder notation: Colon or Vertical Line I remember once hearing offhandedly that in set builder notation, there was a difference between using a colon versus a vertical line, e.g. $\\{x: x \in A\\}$ as opposed to $\\{x\mid x \in A\\}$. I've tried searching for the distinction, but have come up empty-handed.

There is no difference that I've ever heard of. I do strongly prefer "$\vert$" to "$\colon$", though, because I'm often interested in sets of maps, and e.g. $$\\{f \mid f\colon \mathbb{R}\rightarrow\mathbb{C}\text{ with $f(6)=24$}\\}$$ is easier to read than $$\\{f: f\colon \mathbb{R}\rightarrow\mathbb{C}\text{ with $f(6)=24$}\\}$$.

EDIT: Note that as Mike Pierce's answer shows, sometimes "$:$" is clearer. At the end of the day, _use whichever notation is most clear for your context_.