Union of Set of Ordinals is Ordinal

From ProofWiki
Jump to navigation Jump to search


Let $A$ be a set of ordinals.

Then $\bigcup A$ is an ordinal.


Let $y$ be a set.

Let $\On$ be the class of all ordinals.

Let $F: y \to \On$ be a mapping.


$\bigcup \map F y \in \On$

where $\map F y$ is the image of $y$ under $F$.

Proof 1

By Class of All Ordinals is Well-Ordered by Subset Relation, a set of ordinals forms a chain.

The result then follows from Union of Chain of Ordinals is Ordinal.

Proof 2

\(\ds x\) \(\in\) \(\ds \bigcup A\)
\(\ds \leadsto \ \ \) \(\ds \exists y \in A: \, \) \(\ds x\) \(\in\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \exists y \subseteq \bigcup A: \, \) \(\ds x\) \(\subseteq\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\subseteq\) \(\ds \bigcup A\)

From this, we conclude that $\bigcup A$ is a transitive set.

From Class is Transitive iff Union is Subclass, it follows that:

$\bigcup A \subseteq A \subseteq \On$

Note that the above applies as well to sets as it does to classes.

By Subset of Well-Ordered Set is Well-Ordered, $A$ is also well-ordered by $\Epsilon$.

Thus by Alternative Definition of Ordinal, $\bigcup A$ is an ordinal.