[proofplan] We use the standard presentation of $0^\sharp$ as Silver's complete Ehrenfeucht-Mostowski blueprint for $L$. This blueprint produces, for every ordinal order type, a well-founded Ehrenfeucht-Mostowski model whose distinguished generators are indiscernibles. The theorem hypothesis already includes the key Silver fact that the collapsed generators at each order type are order indiscernibles for the full structure $L$. Coherence of the blueprint makes these ordinal generators compatible as the order type varies, and their union is therefore a proper class of ordinals indiscernible for $L$. [/proofplan]

[step:Fix the Silver blueprint coded by $0^\sharp$]By the Silver Ehrenfeucht-Mostowski formulation of $0^\sharp$ stated in the theorem, fix the complete coherent blueprint $\mathcal{E}$ for the Skolemized first-order theory of $L$. Thus, for every ordinal $\theta$, the blueprint $\mathcal{E}$ yields a well-founded extensional Ehrenfeucht-Mostowski model $M_\theta$ generated by a strictly increasing sequence \begin{align*}I_\theta = \{i_{\xi,\theta} : \xi < \theta\}\end{align*} of distinguished generators, and this sequence is a sequence of order indiscernibles in $M_\theta$.[/step]

[guided]The point of using $0^\sharp$ is that it is not merely a set of isolated truths about $L$; in the Silver presentation, it is a coherent Ehrenfeucht-Mostowski blueprint. Let $\mathcal{E}$ denote this blueprint. Its role is to tell us, uniformly for every finite increasing tuple of formal generators, which first-order formulas should hold of that tuple. For each ordinal $\theta$, we apply the Ehrenfeucht-Mostowski construction to the linear order $(\theta,<)$. This gives a structure $M_\theta$ generated by constants indexed by $\theta$: \begin{align*} I_\theta = \{i_{\xi,\theta} : \xi < \theta\}. \end{align*} The defining property of the Ehrenfeucht-Mostowski construction is that the truth of a formula on an increasing tuple of generators depends only on the order type of that tuple. Therefore, if \begin{align*} \xi_1 < \cdots < \xi_n < \theta \end{align*} and \begin{align*} \eta_1 < \cdots < \eta_n < \theta, \end{align*} then for every first-order formula $\varphi(v_1,\dots,v_n)$, \begin{align*} M_\theta \models \varphi(i_{\xi_1,\theta},\dots,i_{\xi_n,\theta}) \iff M_\theta \models \varphi(i_{\eta_1,\theta},\dots,i_{\eta_n,\theta}). \end{align*} The non-routine content supplied by $0^\sharp$ is exactly the Silver Ehrenfeucht-Mostowski package stated in the theorem: well-foundedness, extensionality, collapse to constructible levels, order indiscernibility of the collapsed generator sets for the full structure $L$, and coherence under restriction of order type. In this proof we use that package as the precise formal meaning of the hypothesis that $0^\sharp$ exists in Silver EM form.[/guided]

[step:Collapse the well-founded models into $L$] For each ordinal $\theta$, let \begin{align*}\pi_\theta: M_\theta \to N_\theta\end{align*} denote the Mostowski collapse of $M_\theta$. Since $M_\theta$ is well-founded and extensional, $N_\theta$ is transitive and $\pi_\theta$ is an isomorphism. By the Silver EM data fixed in the theorem statement, the transitive collapse $N_\theta$ is some constructible level $L_{\gamma_\theta}$, for an ordinal $\gamma_\theta$. Define \begin{align*}\alpha_{\xi,\theta} := \pi_\theta(i_{\xi,\theta})\end{align*} for each $\xi < \theta$. Then \begin{align*}A_\theta := \{\alpha_{\xi,\theta} : \xi < \theta\}\end{align*} is a strictly increasing set of ordinals, and by the Silver EM hypothesis $A_\theta$ is a set of order indiscernibles for the full structure $L$. [/step]

[step:Record the finite-stage indiscernibility supplied by the Silver hypothesis] Let $n \in \mathbb{N}$, let $\varphi(v_1,\dots,v_n)$ be a first-order formula in the language of set theory, and let \begin{align*}\xi_1 < \cdots < \xi_n < \theta\end{align*} and \begin{align*}\eta_1 < \cdots < \eta_n < \theta.\end{align*} The Silver EM hypothesis states directly that \begin{align*}A_\theta := \{\alpha_{\xi,\theta} : \xi < \theta\}\end{align*} is a set of order indiscernibles for $L$. Applying that hypothesis to the two increasing tuples from $A_\theta$ gives \begin{align*}L \models \varphi(\alpha_{\xi_1,\theta},\dots,\alpha_{\xi_n,\theta}) \iff L \models \varphi(\alpha_{\eta_1,\theta},\dots,\alpha_{\eta_n,\theta}).\end{align*} Thus each finite-stage set $A_\theta$ is a set of order indiscernibles for $L$ in exactly the sense needed for the final assembly argument. [/step]

[step:Use coherence to assemble one class of indiscernibles] The Silver blueprint $\mathcal{E}$ is coherent under restrictions of order type in the sense stated in the theorem. Therefore, whenever $\theta \leq \theta'$, the model $M_\theta$ is the initial-generator submodel of $M_{\theta'}$ generated by the first $\theta$ distinguished generators, and the collapse maps agree on those generators. Consequently, \begin{align*}\alpha_{\xi,\theta} = \alpha_{\xi,\theta'}\end{align*} for every $\xi < \theta \leq \theta'$. Define a class $I \subset \operatorname{Ord}$ by \begin{align*}I := \{\alpha_\xi : \xi \in \operatorname{Ord}\}\end{align*} where $\alpha_\xi$ denotes the common value of $\alpha_{\xi,\theta}$ for any ordinal $\theta > \xi$. This definition is independent of the choice of $\theta$ by coherence. For every ordinal $\theta$, the sequence $\langle \alpha_\xi : \xi < \theta \rangle$ is strictly increasing because it agrees with the strictly increasing collapsed generator sequence $\langle \alpha_{\xi,\theta} : \xi < \theta \rangle$. Hence $\xi \mapsto \alpha_\xi$ injects $\theta$ into $I$ for every ordinal $\theta$. By the Burali-Forti argument, no set contains injections from all ordinals, so $I$ is a proper class of ordinals. [/step]

[step:Verify that the assembled class is indiscernible for $L$] Let $n \in \mathbb{N}$, let $\varphi(v_1,\dots,v_n)$ be a first-order formula in the language of set theory, and choose increasing tuples \begin{align*}\alpha_{\xi_1} < \cdots < \alpha_{\xi_n}\end{align*} and \begin{align*}\alpha_{\eta_1} < \cdots < \alpha_{\eta_n}\end{align*} from $I$. Choose an ordinal $\theta$ such that \begin{align*}\xi_1,\dots,\xi_n,\eta_1,\dots,\eta_n < \theta.\end{align*} By construction and coherence, these elements of $I$ are exactly the corresponding elements of $A_\theta$. Since $A_\theta$ is a set of order indiscernibles for $L$, we have \begin{align*}L \models \varphi(\alpha_{\xi_1},\dots,\alpha_{\xi_n}) \iff L \models \varphi(\alpha_{\eta_1},\dots,\alpha_{\eta_n}).\end{align*} Thus $I$ is a proper class of order indiscernibles for $L$, as required. [/step]