I've noticed that most of the work in Ackermann set theory is primarily concerned with constructing sets in $V$, the rest of the classes are just excess material, carrying no comprehension over them, there is a try of Muller in which he strengthen the class existence principle of Ackermann into Separation over classes, the resultant theory is $A$, and adding Regularity $R$ and Choice $C$ he gets into $ARC$ a theory claimed [see [here\] ][1]to serve as a foundation of both category and set theory, and thus for most of mathematics.

This gave me the idea of *de-reflection* principle, since Ackermann's set theory can be interpeted in systems using reflection [see [here\] ][2], so if to any of the two systems appearing in that posting, we add the following principle:

**De-Reflection schema:** if $\varphi$ is a sentence in $FOL(=,\in)$, i.e. doesn't use the symbol $V$, and $\varphi^V$ is the bounded by $V$ sentence of $\varphi$, i.e. the sentence obtained by merely bounding every quantifier in $\varphi$ by $V$, then: $ \varphi^V \to \varphi $, is an axiom.

In other words we are reversing the reflection process, so we are concluding things about classes in general by reflecting from the inside of $V$ to outside it. By that, all set axioms (i.e. sentences in the language of set theory that are satisfied in $V$), would generalize over all classes. This way we easily get Muller's theory.

>Question: is there an obvious inconsistency with a theory that both uses reflection and de-reflection principles?


  [1]: http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF
  [2]: https://mathoverflow.net/questions/317658/what-is-the-strength-of-adding-limitation-of-size-and-a-simple-version-of-reflec