Polynomial-time satisfiability tests for boolean fragments of set theory

We recently undertook an investigation aimed at identifying small fragments of set theory (which in most cases are sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator 08 for the time being, in this note we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators 2a, 29, , the Boolean relators 86, / 86, =, /=, and the predicates \u2018\ub7 = 05\u2019 and \u2018Disj(\ub7, \ub7)\u2019, meaning \u2018the argument set is empty\u2019 and \u2018the arguments are disjoint sets\u2019, along with their opposites \u2018\ub7 /= 05\u2019 and \u2018\uacDisj(\ub7, \ub7)\u2019