How to symbolically define set of all real numbers (R) in set-builder notation? Can we define R using set-builder notation without language semantics (purely in math symbols)?
Is it valid to do the following: $\{x\in\mathbb{R}|x\}$
 A: The usual format for describing a set using set-builder notation is:
$$\{\text{what elements of the set look like} \mid \text{what needs to be true of those elements}\}$$
where the writing after the vertical bar is a property (or several properties)  that needs to be true about the object named (or 'instantiated') before the bar.
So, something like $\{x \mid x\in \Bbb R\}$ is more usual. And this just says that our set consists of all things $x$, where $x \in \Bbb R$.
Another example: the set of even integers could be written as $\{2k \mid k \in \Bbb Z\}$. This says that for every integer $k$ (i.e., $k \in \Bbb Z$) we put the integer $2k$ in our set. Or, that our set consists of things of the form $2k$, where $k$ is an integer; all the integers that are multiples of $2$ (the even ones). 
Or, you could write the set of even integers as $\{n \mid \text{$n$ is an even integer}\}$, or $\{n \mid n = 2k \text{ for some } k \in \Bbb Z\}$. There are lots of possibilities.

Your particular example, writing the set of real numbers using set-builder notation, is causing some grief because when you define something, you're essentially creating it out of thin air, possibly with the help of different things. It doesn't really make sense to define a set using the set you're trying to define---and the set of real numbers takes a surprising amount of work to really define adequately, by a mathematician's standards.  
But you don't really care about that, you just want to practice set-builder notation. Hence my suggestion to say that you only want to 'write' or 'denote' the set of real numbers. That sets aside the circularity inherent in defining the set of real numbers, using the set of real numbers. 

No matter what, it's going to look pretty weird writing a set using set-builder notation, using the set we're trying to write. Once you get used to it, you'll see that doing that always looks 'the same', e.g. $\Bbb Z = \{n \mid n \in \Bbb Z\}$.
A: I don't know if you could find a way to do it without any language semantics but you can define $\mathbb{R}$ as the limits of $\mathbb{Q}$-cauchy sequences. 
