13.66 in Switzer's Algebraic Topology: Homotopy and Homology.  The idea is to use the fact that $\Sigma X$ decomposes into two copies of $CX$, say $A$ and $B$, glued along the common boundary of $X$.  For any two cohomology classes $x$ and $y$ in $\tilde{E}^* X$, you can uniquely pull $x$ back to a class $x'$ on the relative pair $(X, A)$ and $y$ back to a class $y'$ on $(X, B)$.  Cupping is natural w.r.t the two relative inclusions $i_A: (X, \{x_0\}) \to (X, A)$ and $i_B: (X, \{x_0\}) \to (X, B)$, and so you get the calculation $x \smile y = i_A^*(x') \smile i_B^*(y') = i^*(x' \smile y')$, where $i: (X, \{x_0\}) \to (X, X)$ is another relative inclusion and $x' \smile y'$ a class on the pair $(X, X)$ --- but that guy has trivial reduced cohomology.