I've seen that the phrase "such that" can be notated as a $|$ or a $:$, specially when you are using a set builder notation. My Geometry teacher also occasionally uses an $\in$ but "backwards" when stating theorems.
But now my Calculus teacher uses a weird symbol that I've never seen before. It's like an normal uppercase E, but "laying down", "rotated -90°", or like three parallel vertical lines with a horizontal line over them, like this:
Does this symbol even exist? Have you seen it before? Where? Is it correct to use it?
Thanks.

I've searched over 900 symbols using this tool and I wasn't able to find such one, nor have I ever seen it used.
My advice would be to just ask your professor if he invented that symbol, and if he did, for what reason.