# No Ordinal Between Set and Successor

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## Theorem

Let $x$ be an ordinal.

Then no ordinal $y$ exists between $x$ and its successor:

- $\neg \exists y: \paren {x \prec y \prec x^+}$

## Proof

Aiming for a contradiction, suppose such an ordinal $y$ exists.

Then, by Ordering on Ordinal is Subset Relation:

- $x \in y$

and from Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:

- $y \in x^+$

Applying the definition of a successor set, we have:

- $y \in x \lor y = x$

But this creates a membership loop, in that:

- $x \in y \in x \lor x \in x$

By No Membership Loops, we have created a contradiction.

The result follows from Proof by Contradiction.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.25$