Yeah in my experience (and on my bookshelf), I see “:”, “|”, “s.t.” And the words “such that” or “with” (this often works in context for lots of properties) and I have literally never seen that inverse membership sign before.

Not a huge fan of quantifier symbols in published notation. For example I use backwards E all over the place in my notes, but in everything I typeset I say “there exists”. Mathematics is supposed to be written in complete sentences anyway so you are going to have some words, so this doesn’t seem a particularly useful two words to turn into a symbol.