In his Denumerable models of complete theories, 1961, Thm. 3.4., Vaught is said to have proven that $\mathfrak{A}$ is a prime model of the theory $Th(\mathfrak{A})$ if every element of $\mathfrak{A}$ is definable.
Unfortunately, I don't have access to this article. Can somebody either tell me how Vaught did it, give me another reference to a proof (perhaps in some textbook), or sketch a proof here?
Perhaps it is as simple as the following?
Assume that $\mathfrak{B}\equiv\mathfrak{A}$, and assume each element of $\mathfrak{A}$ is definable. Then for each $a\in A$, let $\phi_a$ be such that $\mathfrak{A}\models \phi_a(a)$ and for all $c\in A\setminus\{a\}$, $\mathfrak{A}\not\models \phi_a(c)$. Since for each $a$, $\mathfrak{B}\models \exists x \phi_a(x)$, send $a$ to a witness $b_a\in B$, i.e., $\mathfrak{B}\models \phi_a(b_a)$. Then verify that the mapping $f:a\mapsto b_a$ is an elementary embedding.
Or am I missing something?