Let $\mathcal{L}$ be a countable propositional language, and let $\vdash_{\mathrm{IPC}}$ denote intuitionistic propositional derivability. If $\Gamma$ is a set of $\mathcal{L}$-formulas and $A$ is an $\mathcal{L}$-formula with $\Gamma \nvdash_{\mathrm{IPC}} A$, then there exists a set $\Delta$ of $\mathcal{L}$-formulas such that $\Gamma \subseteq \Delta$, $A \notin \Delta$, $\Delta$ is deductively closed under $\vdash_{\mathrm{IPC}}$, and $\Delta$ is disjunction-prime: whenever $B \lor C \in \Delta$, either $B \in \Delta$ or $C \in \Delta$.