Noetherian local ring of positive dimension has element in maximal ideal outside minimal primes

Statement
Suppose $$(A,\mathfrak{m})$$ is a Noetherian local ring that is not Artinian: in other words, the unique maximal ideal $$\mathfrak{m}$$ is not a minimal prime. Then, there exists an element $$x \in \mathfrak{m}$$ such that $$x$$ is not in any minimal prime.

For a reduced ring, i.e. a ring with no nilpotents, it is also clearly true that any element outside the union of minimal primes must be a nonzerodivisor. While this may not be true in general for arbitrary rings, the effect as far as Hilbert-Samuel polynomials goes, is the same as for a nonzerodivisor.

Facts used

 * In a Noetherian ring, every prime contains a minimal prime, and there are only finitely many minimal primes
 * Prime avoidance lemma