Union of Successor Ordinal

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Theorem

Let $x$ be an ordinal.

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

Then:

$\bigcup \left({x^+}\right) = x$


Proof

\(\displaystyle \bigcup \left({x^+}\right)\) \(=\) \(\displaystyle \bigcup \left({x \cup \left\{ {x}\right\} }\right)\) Definition of Successor Set
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup x \cup \bigcup \left\{ {x}\right\} }\right)\) Union Distributes over Union/Sets of Sets
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup x \cup x}\right)\) Union of Singleton
\(\displaystyle \) \(=\) \(\displaystyle x\) Class is Transitive iff Union is Subset

$\blacksquare$