Very simple question Are the Pairing Axiom and the Union axiom in the Zermelo–Fraenkel set theory the same? I have a book that states them as the following: Pairing Axiom: For any sets u and v, there is a set having as members just u and v. Union axiom: For any sets a and b there exists a set whose members are those belonging to either a or b. Also in the book, they give these definitions in the form of a logic definition (I'd post but I can't find some of the symbols in any LaTex reference), the definitions are completely identical. So are they different and if so what is the difference (and what would I be able to prove with one but not the other). Thanks!