Ordinal is Subset of Successor

Theorem
Let $x$ and $y$ be ordinals. Let $x^+$ denote the successor of $x$.

Then, $x \subseteq y^+ \iff ( x \subseteq y \lor x = y^+ )$.

Proof
We shall use $\subset$, denoting proper subset and $\in$, denoting membership, interchangeably. This is justified by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

Conversely: