Associated primes turns short exact sequences to sub-unions

Statement
Suppose $$A$$ is a commutative unital ring. Consider a short exact sequence of modules:

$$0 \to M \to N \to L \to 0$$

Then we have:

$$Ass_A(N) \subset Ass_A(M) \cup Ass_A(L)$$

where $$Ass_A$$ denotes the set of associated primes.

Proof outline
The key nontrivial ingredient, where we actually use primeness, is the following fact:


 * Principal ideal is isomorphic to integral domain as a module: Any principal ideal in an integral domain is isomorphic to the integral domain as a module. Another way of putting this is that any submodule of an integral domain, contains a submodule isomorphic to the whole domain (as a module).