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

Conversely:

\(\displaystyle x \subseteq y^+\) \(\implies\) \(\displaystyle \left({x \subset y^+ \lor x = y^+}\right)\)
\(\displaystyle x \subset y^+\) \(\implies\) \(\displaystyle x \in y^+\) 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

$\blacksquare$