Axioms needed to prove existence of unordered triplet

Question

I know the defenition for ordered triple: just define $\\(a,b)$ to be $\{\{a\},\{a,b\}\}$ and apply this reasoning to $\\((a,b),c)$. But how about unordered triple? And what axioms guarantee us that such triple exists?

Answer 1

For $\mathsf {ZFC}$ we use Pair axiom and Union axiom:

$\{ x,y,z \} = \{ x,y \} \cup \{ z \}$.