How to represent "not an empty set"?

It is perfectly fine to write $|A|>0$. However, the simplest and most common way to write this in symbols would be $$A\neq\emptyset.$$ Note that you don't want to write $|A|\neq \emptyset$, as it is $A$ itself which you are saying is not the empty set, rather than the cardinality of $A$.

(The standard symbol in mathematics for "not equal" is $\neq$, rather than $!{=}$. You can make this symbol in $\LaTeX$ with the command \neq.)

As mentioned in user21820's nice answer below, though, it is also very common to just write this in words ("$A$ is not empty" or "$A$ is nonempty") instead of symbols.

None of the other answers so far mention that professional mathematicians don't specially go out of the way to convert everything to symbols. "$A$ is non-empty" is indeed the most common way to express the statement. Furthermore, for complicated structures it is almost always expressed this way, such as:

Given any non-empty chain of fields ordered by inclusion, their union is also a field.

$A \neq \varnothing$

[LaTeX: A \neq \varnothing]