I am studying enderton elements of set theory and trying to figure out what combinations of axioms are redundant. Given: extensionality, empty set, pair set, union, power set, SUBSETS, AOC, replacement, infinity, regularity. My most pertinent question: I know I can delete subset and pairing. Can I instead delete subset and power set?
No, the power set axiom adds a lot of strength to the theory. In fact, without it, you cannot even prove there are uncountable sets. The set of hereditarily countable sets is a model of ZFC minus power set.