How can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of elementary functions?

The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.

EDIT: More formally, by definition an elementary function is obtained from complex constants and the variable $x$ by a finite number of steps of the following forms:

1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 \ne 0$) $f_1/f_2$ are elementary.
2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.
3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $\log g$ are elementary).

To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose the result is true for elementary functions obtained in at most $n$ steps. If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.