Successor in Limit Ordinal

Theorem
Let $x$ be a limit ordinal.

Let $y \in x$.

Then $y^+ \in x$ where $y^+$ denotes the successor set of $y$:


 * $\forall y \in x: y^+ \in x$

Proof
Because $x$ is a limit ordinal:
 * $x \ne y^+$

Moreover, by Successor of Element of Ordinal is Subset:
 * $y \in x \implies y^+ \subseteq x$

Therefore by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:
 * $y^+ \subset x$ and $y^+ \in x$