How do the power-set axiom and union axiom construct new sets in Zermelo-Fraenkel set theory?

In Zermelo-Fraenkel (ZF) set theory, the power-set axiom and the union axiom serve as fundamental tools for constructing new sets from existing ones, embodying the iterative conception of set formation.1

• Power-set axiom: This axiom states that for any given set $S$, there exists a set $S^1$ that contains all and only the subsets of $S$. By taking the power set of a set, one effectively moves to a higher level in the iterative hierarchy, as the new set contains all possible groupings of the original set's elements.12
• Union axiom: This axiom asserts that if $S$ is a set of sets, there exists a set containing all and only the members of the sets contained in $S$. Essentially, it "flattens" the structure by collecting all elements from the inner sets into a single, unified set.1

Together, these operations allow for the construction of an iterative hierarchy of sets, starting from the empty set and extending through transfinite levels by repeatedly applying power set and union operations.2