An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ such that $\xi < \beta$. As such $|\xi|$ is smaller than that of $\beta$, and thus it is smaller than that of $\alpha$.
Does this sound correct/in the right direction? If so, could someone help better explain the first line?