Fundamental theorem of stability

For a complete theory, the following are equivalent: the theory is stable; no formula has the order property; and every complete type over every model is definable.

A type $p(x)$ over a model $M$ is definable when, for every formula $\phi (x;y)$, there is an $M$-formula $d_p\phi (y)$ such thatfor every tuple $b$ from $M$.

If $T$ is stable, $M\preccurlyeq N\models T$, and $X\subseteq N^n$ is definable with parameters from $N$, then $X\cap M^n$ is definable with parameters from $M$. Apply definability of the type over $M$ of the parameter defining $X$.