# 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

$\blacksquare$