What makes a condition unary vs. n-ary (n>1)? For any two disjoint sets $A$ and $B$, a set $W$ is a connection of $A$ with $B$ if


*

*$Z\in W\implies (\exists x\in A)(\exists y\in B)[Z=\{x,y\}]$

*$(\forall x\in A)(\exists !y\in B)[\{x,y\}\in W]$

*$(\forall y\in B)(\exists !x\in A)[\{x,y\}\in W]$


I know that each of the conditions 1-3 are definite conditions (and indeed being a connection in general is a definite condition). But I'm having trouble telling when I should consider something an $n$-ary condition $(n>1)$ vs a unary condition.
I want to use the Axiom of Separation (Zermelo axioms) which requires a definite condition to be unary. The condition that I want to use is that of a set being a connection. But, to me, that would be a ternary condition, say $P(A,B,W)$. So I would have to "decompose" this ternary condition into unary conditions and apply the separation axiom multiple times, no?

Say, $P(A,B,W)$ is the first order proposition that expresses that $W$ is a connection between $A$ and $B$, Then consider
$${\rm isConn}(W):= " \exists A\exists B:P(A,B,W)"\,.$$
