Union of Successor Ordinal

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Theorem

Let $x$ be an ordinal.

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

Then:

$\displaystyle \map \bigcup {x^+} = x$


Proof

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

$\blacksquare$