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Hamkins and Kikuchi (2016 and 2017) show that in both set theory and classtheory the definable subset ordering of the universe interprets a complete anddecidable theory. If $\mathcal{M}$ is a model of set theory, then $\langle M,\subseteq^\mathcal{M} \rangle$ is an atomic unbounded relatively complementeddistributive lattice. If $\mathcal{M}$ is model of class theory, then $\langleM, \subseteq^\mathcal{M} \rangle$ is an infinite atomic boolean algebra. Weidentify the minimal subsystem of $\mathrm{ZF}$, $\mathrm{BAS}$, that ensuresthat the definable subset relation is an atomic unbounded relativelycomplemented distributive lattice and classify the atomic unbounded relativelycomplemented distributive lattices that can be realised as a subset relationsof this theory. The fact that the theory of atomic unbounded relativelycomplemented distributive lattices is complete is used to show that$\mathrm{BAS}$ decides every $2$-stratified sentence of set theory. We alsoidentify the minimal subsystem of $\mathrm{NBG}$, $\mathrm{BAC}$, that ensuresthat the definable subset relation is an infinite atomic Boolean algebra. Weshow that there is a complete extension, $\mathrm{IABA}_\mathrm{Ideal}$, of thetheory of infinite atomic boolean algebras and an extension $\mathrm{BAC}^+$ of$\mathrm{BAC}$ corresponding to the minimal theory such that if $\mathcal{M}$is a model of $\mathrm{BAC}^+$, then $\langle M, \mathcal{S}^\mathcal{M},\subseteq^\mathcal{M} \rangle$ is a model of $\mathrm{IABA}_{\mathrm{Ideal}}$,where $\mathcal{S}^\mathcal{M}$ is a unary predicate that distinguishes setsfrom classes. This is used to show that $\mathrm{BAC}^+$, a subsystem of$\mathrm{NBG}$, decides every $2$-sentence in the language of class theory thatincludes a unary predicate distinguishing sets from classes.

The subset relation and $2$-stratified sentences in set theory and class theory