It is proved that the axiom schemes for \textit{H. L. Skala's} set theory [Z. Math. Logik Grundlagen Math. 20, 233-237 (1974; Zbl 0301.02072)] are equivalent to the existence of the union and the intersection of all sets satisfying an arbitrary predicate. These axiom schemes of General Union (GU) and Intersection (GI), in conjunction with the axiom of extensionality, define a double Heyting algebra called UI whose complemented elements form an atomic Boolean algebra. The relation L of conditional membership defined by \(L(a,b)\leftrightarrow \forall x(a\in x\to b\in x)\) is not trivial in UI. Using L, the following methodological justification for UI is given (Theorem 3). If, in predicate logic extended by \(\in\), some predicate P(x) is comprehensible, then \(P(a)\wedge L(a,b)\to P(b)\) is valid. Postulating that this necessary condition is sufficient for comprehensibility yields \((GU)+(GI)\). An aspect of the categorical (non-modal) definability of modalities is elucidated by the proposed interpretation of the two ''modes'' of a predicate as introduced by Skala. In UI, these modes are the union of all sets whose elements all satisfy the predicate and the intersection of all sets containing every set satisfying the predicate as element. No strengthening of UI is carried out but possible extensions are being discussed. (UI \(+\) symmetry of L) gives the consistent Skala set theory proper, incompatible with (UI \(+\) antisymmetry of L) securing an infinite universe. After a few manipulations, not only well-orderings of order type \(\alpha +2\) and transitive power sets, but also the impredicative BG set theory (MK) become models of UI. Thus UI can easily be extended to a theory consistent with, and roughly equivalent to, MK. The comprehension schemes of ZF, MK, UI, and Skala's theory proper are shown to be all of the form \(\exists y\forall x(x\in y\leftrightarrow \exists z(P(z)\wedge R(x,z)))\) with a characteristic relation R for every system.