Foundational Relation has no Relational Loops

Theorem

Let $\prec$ be a foundational relation on $A$ and let $x_1, x_2, \ldots, x_n \in A$.

Then:

$\neg \left({x_1 \prec x_2 \land x_3 \prec x_4 \cdots \land x_n \prec x_1}\right)$

That is, there are no relational loops within $A$.

Proof

Since $x_1, x_2, \ldots, x_n \in A$, there exists a $y$ such that $y = \left\{{x_1, x_2, \ldots, x_n}\right\}$.

Then $y$ is a nonempty subset of $A$.

So, by the definition of a foundational relation:

$\exists w \in y: \forall z \in y: \neg w \prec z$

Now, suppose $x_1 \prec x_2 \land x_3 \prec x_4 \cdots \land x_n \prec x_1$.

But since the elements of $y$ are $x_1, x_2, \ldots, x_n$, then this contradicts the previous statement, since:

$\forall w \in y: \exists z \in y: w \prec z$

Thus, a founded relation has no relational loops.

$\blacksquare$

Also see

• Foundational Relation
• Axiom of Foundation

• Epsilon is Foundational

Source

• Gaisi Takeuti: Introduction to Axiomatic Set Theory (1971): $6.23$