# Foundational Relation has no Relational Loops

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## Contents

## 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 $x_1 \prec x_2 \land x_2 \prec x_3 \cdots \land x_n \prec x_1$.

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

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $6.23$