# No Membership Loops

 It has been suggested that this article or section be renamed: Something that indicates the domain on which the definition applies. One may discuss this suggestion on the talk page.

## Theorem

For any proper classes or sets $A_1, A_2, \ldots, A_n$:

$\neg \left({A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}\right)$

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

Either $A_1, A_2, \ldots, A_n$ are all sets, or there exists a proper class $A_m$ such that $1 \le m \le n$.

Suppose there exists a proper class $A_m$.

Then, by the definition of a proper class, $\neg A_m \in A_{m+1}$, since it is not a member of any class.

The result then follows directly.

Otherwise it follows that all $A_1, A_2, \ldots, A_n$ are sets.

Then, by the fact that Epsilon is Foundational and a Foundational Relation has no Relational Loops, it follows that:

$\neg \left({A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}\right)$

$\blacksquare$