No Natural Number between Number and Successor

Theorem
Let $x$ be an ordinal. Then, no ordinal exists between $x$ and its successor.


 * $\displaystyle \neg x \prec y \prec x^+$

Proof
We will proceed by contradiction. Assume such a set exists. Then, by Ordering on an Ordinal is Subset Relation and Ordinal Proper Subset Membership, $x \in y \land y \in x^+$.

Applying the definition of a successor set, $y \in x \lor y = x$. But this creates a membership loop, because $x \in y \in x \lor x \in x$. By No Membership Loops, we have created a contradiction.