I am trying to show (if possible) that symmetric submodular functions are non-monotone (excluding constant sub-modular functions).

Recall that a submodular function $f : 2^{\Omega} \rightarrow R$ is such that for $A,B \subset \Omega$, $f(A)+f(B) \geq f(A\cup B) + f(A \cap B)$, and that a symmetric submodular function satisfies $f(A) = f(\Omega \setminus A)$. A monotone function satisfies $f(A) \leq f(B)$ whenever $A \subseteq B$.

A classic example of a symmetric submodular function that is non-monotone is the graph cut function.

Is it possible to show just from the above definitions that any symmetric submodular function (that is not constant) is non-monotone?