[proofplan] We enumerate the countable model $M$ and build, by induction, finite partial elementary maps into an arbitrary model $N \models T$. Atomicity supplies an isolating formula for each finite tuple from $M$. The key extension step is that an isolating formula for the already-mapped prefix transfers the existential assertion needed to realize the next isolated type in $N$. The union of the finite maps then preserves every formula on every finite tuple, hence is an elementary embedding $M \to N$. [/proofplan]

[step:Enumerate the model and fix the target model] Let $N \models T$ be arbitrary. Since $M$ is countable, choose an enumeration \begin{align*} M = \{a_k : k \in \mathbb{N}\} \end{align*} without repetitions. For each $k \in \mathbb{N}$, define the finite tuple \begin{align*} \bar{a}_k := (a_1,\dots,a_k) \in M^k. \end{align*} Because $M$ is atomic, the complete type $\operatorname{tp}^M(\bar{a}_k/\varnothing)$ is isolated. Hence for each $k \in \mathbb{N}$ choose an $\mathcal{L}$-formula \begin{align*} \theta_k(x_1,\dots,x_k) \end{align*} which isolates $\operatorname{tp}^M(\bar{a}_k/\varnothing)$, meaning that $M \models \theta_k(\bar{a}_k)$ and, for every $\mathcal{L}$-formula $\psi(x_1,\dots,x_k)$, \begin{align*} M \models \psi(\bar{a}_k) \quad\Longleftrightarrow\quad T \models \forall x_1\dots \forall x_k\bigl(\theta_k(x_1,\dots,x_k) \to \psi(x_1,\dots,x_k)\bigr). \end{align*} [/step]

[step:Build finite tuples in $N$ realizing the corresponding isolated types]We construct, by induction on $k \in \mathbb{N}$, tuples \begin{align*} \bar{b}_k := (b_1,\dots,b_k) \in N^k \end{align*} such that \begin{align*} N \models \theta_k(\bar{b}_k). \end{align*} For $k = 1$, since $M \models \exists x_1\,\theta_1(x_1)$ and $M \models T$, completeness of $T$ gives \begin{align*} T \models \exists x_1\,\theta_1(x_1). \end{align*} As $N \models T$, choose $b_1 \in N$ such that $N \models \theta_1(b_1)$. Assume now that $k \in \mathbb{N}$ and that $\bar{b}_k \in N^k$ has been chosen with $N \models \theta_k(\bar{b}_k)$. Define the $\mathcal{L}$-formula \begin{align*} \exists x_{k+1}\,\theta_{k+1}(x_1,\dots,x_k,x_{k+1}) \end{align*} in the free variables $x_1,\dots,x_k$. Since $M \models \theta_{k+1}(\bar{a}_{k+1})$, we have \begin{align*} M \models \exists x_{k+1}\,\theta_{k+1}(\bar{a}_k,x_{k+1}). \end{align*} Because $\theta_k$ isolates $\operatorname{tp}^M(\bar{a}_k/\varnothing)$, it follows that \begin{align*} T \models \forall x_1\dots \forall x_k \left( \theta_k(x_1,\dots,x_k) \to \exists x_{k+1}\,\theta_{k+1}(x_1,\dots,x_k,x_{k+1}) \right). \end{align*} Since $N \models T$ and $N \models \theta_k(\bar{b}_k)$, there exists $b_{k+1} \in N$ such that \begin{align*} N \models \theta_{k+1}(\bar{b}_k,b_{k+1}). \end{align*} This completes the induction.[/step]

[guided]The purpose of the induction is to choose images for the elements of $M$ one at a time, while ensuring that every finite initial tuple has exactly the same complete type in $N$ as the corresponding tuple in $M$. For the first element, atomicity gives an isolating formula $\theta_1(x_1)$ for $\operatorname{tp}^M(a_1/\varnothing)$. Since $M \models \theta_1(a_1)$, the sentence $\exists x_1\,\theta_1(x_1)$ is true in $M$. Because $T$ is complete and $M \models T$, every sentence true in $M$ is a consequence of $T$. Thus \begin{align*} T \models \exists x_1\,\theta_1(x_1). \end{align*} Since $N \models T$, there is $b_1 \in N$ with \begin{align*} N \models \theta_1(b_1). \end{align*} Now suppose we have already chosen $\bar{b}_k = (b_1,\dots,b_k) \in N^k$ such that \begin{align*} N \models \theta_k(\bar{b}_k). \end{align*} We need to choose $b_{k+1}$ so that the longer tuple realizes the isolated type of $(a_1,\dots,a_k,a_{k+1})$. The formula $\theta_{k+1}(x_1,\dots,x_k,x_{k+1})$ isolates that longer type, and $M$ realizes it at $\bar{a}_{k+1}$. Therefore \begin{align*} M \models \exists x_{k+1}\,\theta_{k+1}(\bar{a}_k,x_{k+1}). \end{align*} The important point is that this existential statement depends only on the complete type of the prefix $\bar{a}_k$. Since $\theta_k$ isolates $\operatorname{tp}^M(\bar{a}_k/\varnothing)$, every formula true of $\bar{a}_k$ is forced by $T$ together with $\theta_k$. Applying this to the formula \begin{align*} \exists x_{k+1}\,\theta_{k+1}(x_1,\dots,x_k,x_{k+1}), \end{align*} we obtain \begin{align*} T \models \forall x_1\dots \forall x_k \left( \theta_k(x_1,\dots,x_k) \to \exists x_{k+1}\,\theta_{k+1}(x_1,\dots,x_k,x_{k+1}) \right). \end{align*} Since $N \models T$ and $\bar{b}_k$ realizes $\theta_k$, the displayed implication holds in $N$ at $\bar{b}_k$. Hence there exists $b_{k+1} \in N$ satisfying \begin{align*} N \models \theta_{k+1}(\bar{b}_k,b_{k+1}). \end{align*} This is exactly the extension needed for the induction.[/guided]

[step:Define the limiting map and prove that it preserves all formulas] Define the map \begin{align*} f: M &\to N \\ a_k &\mapsto b_k. \end{align*} This is well-defined because the enumeration of $M$ has no repetitions. Let $m \in \mathbb{N}$, let $i_1,\dots,i_m \in \mathbb{N}$, and let $\varphi(y_1,\dots,y_m)$ be an $\mathcal{L}$-formula. Choose $k \in \mathbb{N}$ such that $i_1,\dots,i_m \leq k$. Define the $\mathcal{L}$-formula \begin{align*} \psi(x_1,\dots,x_k) := \varphi(x_{i_1},\dots,x_{i_m}). \end{align*} Since $\theta_k$ isolates $\operatorname{tp}^M(\bar{a}_k/\varnothing)$ and $N \models \theta_k(\bar{b}_k)$, the tuples $\bar{a}_k$ and $\bar{b}_k$ satisfy the same $\mathcal{L}$-formulas. Therefore \begin{align*} M \models \psi(\bar{a}_k) \quad\Longleftrightarrow\quad N \models \psi(\bar{b}_k). \end{align*} By the definition of $\psi$, this is precisely \begin{align*} M \models \varphi(a_{i_1},\dots,a_{i_m}) \quad\Longleftrightarrow\quad N \models \varphi(f(a_{i_1}),\dots,f(a_{i_m})). \end{align*} Thus $f$ preserves and reflects every $\mathcal{L}$-formula on every finite tuple from $M$. Taking $\varphi(y_1,y_2)$ to be $y_1 = y_2$ shows that $f$ is injective. Since $f$ preserves and reflects every formula, it is an elementary embedding $M \to N$. [/step]

[step:Conclude primeness] The model $N \models T$ was arbitrary. We have shown that for every model $N \models T$ there exists an elementary embedding \begin{align*} f: M \to N. \end{align*} Therefore $M$ is prime. [/step]