Anyone familiar with a mathematical symbol for "that is" / "to clarify" / "whereby"? Many years ago, a math teacher of mine thought us that there was a mathematical symbol for "that is" / "to clarify" / "whereby" (I'm not from an English speaking country, so I'm writing several alternatives here to try to approximate the actual meaning that I was thought), which looked a bit like: "$\supset :$" (or rather, a backwards capital C (alternatively an open o; the symbol was handwritten, so hard to tell) followed by a colon, but I don't know how to write that here).
As for usage, my teacher used it when doing algebra when she wanted to skip a few lines (hence, I suppose you could take it to mean "one sees easily that", but that's not how she "translated" the symbol), as in:
\begin{align}
&6x + 2 = 20 \\
\supset: \quad &x= 3,
\end{align}
(this being an overly simple example), or when she wanted to re-state the conclusion of an algebraic derivation using slightly different notation which better reflected whatever it was that she wanted us to take away from said derivation, as in:
\begin{align}
&6x + 2 = 20 \\
&\quad \vdots \\
& x = 3 \\
\supset: \quad &x= \sqrt{9},
\end{align}
(this still being an overly simple example, and possibly a somewhat silly one).
However, I haven't been able to find anything on the web confirming the existence of this symbol, and my question is therefore: Have anyone else seen anything like this?
As for the the necessity of this symbol, I do realize that this symbol is rather redundant, since one might as well use "$\implies$", "$\iff$", simply some text, or possibly (in some cases) even "$\therefore$".
 A: It's not a symbol, but the abbreviation "i.e." is short for the Latin phrase id est which is typically translated as "that is." The abbreviation is very commonly used for this meaning in math. 
A: The symbol $\supset$ is sometimes used in propositional logic (in particular in philosophical propositional logic) to mean material conditional.
Thus $p \supset q$ is another notation for the more common $p \rightarrow q$ and means if $p$ is true, then $q$ is also true.
I let you read the Wikipedia entry material conditional for a precise definition and alternative notation.
As for "$\supset:$", I just know that some logicians add sporadic symbols ":" in their formulas, with no real logical meaning.
A: Please don't use such symbols in your mathematical writing. Instead, explain these part of your mathematical ideas in words. Strive to write all your mathematical ideas in complete sentences, with mathematical symbolism and notation embedded within those sentences. A mathematical argument or even a computation can be thought of as a kind of essay, and it will be clearer for you to write it that way. It is especially important to include the logical cue words, such as "because", "consequently," "therefore," "in light of the fact that," "by definition," and so on, which explain the connection between your mathematical assertions and how the reader is to understand them.
For example, don't use three dots in a triangle as some do for "therefore" or a different arrangement of three dots for "because." We already have perfectly adequate and universally understood words in our language to express those ideas; no extra notation is required. Don't try to communicate complicated ideas mainly with arrows stretching across the page. Rather, explain your argument in ordinary language; your readers will learn far more. 
Some people go a bit further, and recommend that one should not use notation like $\exists$ and $\forall$ or $\implies$ and $\iff$ in mathematical exposition, unless one is specifically talking about a formal assertion as such. Instead, one should write out "if and only if" or "if, ...then" or whatever it is that is meant. The symbols are to be used only when specifically mentioning the corresponding relation rather than using it, as in the use/mention distinction. The symbols can be used for especially complicated assertions or to refer to a particular formal assertion. 
A general rule of thumb: For good mathematical writing, use mathematical symbols and notation only when your mathematical idea cannot be adequately expressed without them. Simplify or eliminate your notation as much as possible.
I recommend that you explain your mathematical ideas as much as possible in ordinary natural language, even if this takes more space. 
I make this advice for essentially all mathematical writing, including not only formal papers, but also homework solutions and writing mathematics on a chalkboard. I would be a little more relaxed concerning one's own lecture notes and scratch work and so on, except for the fact that it simply doesn't take appreciably longer to write in a clear manner even in informal cases.
