A formula that maps elements to proper classes in an arbitrary $\in$-universe

Abstract

In this paper we construct a formula $\varphi(x,a)$ on the language of set theory $\mathcal{L}\mathbin{:}(\in,=)$ such that $\{x\ | \ \varphi(x,a)\}$ is a proper class for each element $a$ and such that if $a\neq a'$, the classes $\{x\ | \ \varphi(x,a)\}$ and $\{x\ | \ \varphi(x,a')\}$ are different. This formula works for any structure with the exception of 2 structures with two elements each. This formula ``maps'' elements to proper classes injectively.