Downward Lowenheim-Skolem theorem

If $A$ is an infinite $L$-structure and $|L|+{\aleph }_0\le \kappa \le |A|$, then $A$ has an elementary substructure of cardinality $\kappa$. One proof adds Skolem functions for existential formulas and closes a chosen $\kappa$-element subset under them; the closure still has size $\kappa$, and the Tarski--Vaught test makes it elementary.