Relations vs. Sets of Ordered Pairs

(Normally in math) a relation is defined to be a sets of ordered pairs.

But the `elementhood' relation between sets can't, itself, be a set of ordered pairs - since there can't be a set which contains each ordered pair of sets such that x is an element of y. [From the existence of such a set you could use the axiom of collection in ZF to derive the existence of a set of all sets, and hence the Russell set and contradiction.]

Therefore, not all relations (in the ordinary sense) are sets of ordered pairs (i.e. relations in the mathematical sense).