Working in language $\mathcal L_{\Omega+, \Omega+}$$\mathcal L_{\Omega^+,\Omega^+}$ where $\Omega$ is the first strongly inaccessible cardinal. If we add a primitive partial $\Omega$-ary function $F$ and a primitive constant $\varnothing$, then axiomatize:
Exclusion: $(\forall x_i)_{i < \Omega}: F((x_i)_{i < \Omega}) \neq \varnothing$
Commutative:$(\forall x_i)_{i < \Omega} (\forall y_i)_{i < \Omega} : \\F((x_i)_{i < \Omega}) =F((y_i)_{i < \Omega}) \iff \bigwedge_{i < \Omega} (\bigvee_{j< \Omega} (x_i = y_j)) \land \bigwedge_{j < \Omega} (\bigvee_{i < \Omega} (x_i = y_j))$
Existence:$ (\forall x_i)_{i < \Omega}: \\ \bigvee_{1+\kappa < \Omega}[(\exists y_j)_{j < 1+\kappa} (\bigwedge_{i < \Omega} (\bigvee_{j < 1+\kappa} (x_i=y_j)))] \iff \exists y: y=F((x_i)_{i < \Omega })$
Restriction: $\forall x: x=\varnothing \lor (\exists x_i)_{i < \Omega}: y=F((x_i)_{i < \Omega})$
If we define set membership $\in$ as:
Define: $a \in b \iff (\exists x_i)_{i < \Omega}: b=F((x_i) _{i < \Omega}) \land \bigvee_{i < \Omega} (a=x_i)$
Would that interpret $\sf ZFC$-$\sf Reg.$?