No Membership Loops

Theorem
For any proper classes or sets $A _1, A _2,... A _n$,

$\neg ( A _1 \in A _2 \land A _2 \in A _3 ... \land A _n \in A _1 )$

Proof
Either $A _1, A _2,... A _n$ are all sets, or there exists a proper class $A _m$.

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.

Suppose, on the other hand, that all $A _1, A _2,... A _n$ are sets. Then, by the fact that Epsilon is Foundational and a Foundational Relation has no Relational Loops, we can thus infer that $\neg ( A _1 \in A _2 \land A _2 \in A _3 ... \land A _n \in A _1 )$ {qed}

Source

 * : $5.20$