I find this in my Set theory material:

[0] = {x:0==x(mod2)} = {x:2|0-x}


, where I'm replacing equivalence sign ("=" with extra horizontal line) with double equality sign "==" because I don't know how to input the proper symbol.

class 0 includes (all x that are defined by 0 being equivalent to x%2) which is equal to (all x that are defined by 2 being such that 0-x)


{x:2|0-x}


Shouldn't there be be a boolean statement to the right of "|"? Or am I misreading?

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$|$ in this case means 'divides': https://en.wikipedia.org/wiki/Divisibility#Definition
The usual convention for defining a class is this $$\Bigl\{\textbf{variable }\Large\text{separator }\normalsize\textbf{condition on the variable}\Bigr\}$$
The separator can be $:$ (colon) or it can be $\mid$ (pipe). In this case, the symbol $\mid$ is already used to denote the divisibility relation, therefore $:$ was used.