# Ordinal is Subset of Successor

## Theorem

Let $x$ and $y$ be ordinals.

Let $x^+$ denote the successor of $x$.

Then:

$x \subseteq y^+ \iff \left({x \subseteq y \lor x = y^+}\right)$

## Proof

Let $A \subset B$ denote that $A$ is a proper subset of $B$.

Let $A \in B$ denote that $A$ is an element of $B$.

From Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, $\subset$ and $\in$ can be used interchangeably.

Thus:

 $\displaystyle x \subseteq y$ $\implies$ $\displaystyle x \subseteq y^+$ Ordinal is Less than Successor $\displaystyle x = y^+$ $\implies$ $\displaystyle x \subseteq y^+$ Definition 2 of Set Equality
 $\displaystyle x \subseteq y^+$ $\implies$ $\displaystyle \left({x \subset y^+ \lor x = y^+}\right)$ $\displaystyle x \subset y^+$ $\implies$ $\displaystyle x \in y^+$ Transitive Set is Proper Subset of Ordinal iff Element of Ordinal $\displaystyle$ $\implies$ $\displaystyle \left({x = y \lor x \in y}\right)$ Definition of Successor Set $\displaystyle$ $\implies$ $\displaystyle x \subseteq y$ Transitive Set is Proper Subset of Ordinal iff Element of Ordinal

$\blacksquare$