# Successor of Element of Ordinal is Subset

## Theorem

Let $x$ and $y$ be ordinals.

Then:

$x \in y \iff x^+ \subseteq y$

## Proof

 $\displaystyle x \in y$ $\iff$ $\displaystyle x^+ \in y^+$ Successor is Less than Successor $\displaystyle$ $\iff$ $\displaystyle \left({ x^+ \in y \lor x^+ = y }\right)$ definition of successor set $\displaystyle$ $\iff$ $\displaystyle \left({ x^+ \subset y \lor x^+ = y }\right)$ Transitive Set is Proper Subset of Ordinal iff Element of Ordinal $\displaystyle$ $\iff$ $\displaystyle x^+ \subseteq y$

$\blacksquare$