Successor of Element of Ordinal is Subset

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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
\(\displaystyle \) \(\iff\) \(\displaystyle x^+ \subseteq y\)

$\blacksquare$