Union of Successor Ordinal

Theorem

Let $x$ be an ordinal.

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

Then:

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

Proof

 $\ds \map \bigcup {x^+}$ $=$ $\ds \map \bigcup {x \cup \set x}$ Definition of Successor Set $\ds$ $=$ $\ds \paren {\bigcup x \cup \bigcup \set x}$ Set Union is Self-Distributive/Sets of Sets $\ds$ $=$ $\ds \paren {\bigcup x \cup x}$ Union of Singleton $\ds$ $=$ $\ds x$ Class is Transitive iff Union is Subclass

$\blacksquare$