# Ordinal is Subset of Successor

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## 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 |

$\blacksquare$