No Natural Number between Number and Successor

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
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.