Axiom of Pairing Axiom of Pairing states that if $a,b$ are sets, $\exists$ a set $A$ such that $A=\{{a,b\}}$. My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?

User wrote:

My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?

To apply Specification to construct $A=\{a,b\}$, $a$ and $b$ would have to have been assumed or proven to be elements of some other set $B$, i.e. $a\in B$ and $b\in B$. Then $A=\{x|x\in B \land [x=a \vee x=b]\}$. Pairing has no such requirement.
