Successor is Less than Successor

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Theorem

Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$.

Then, $x \in y \iff x^+ \in y^+$.

Proof

\(\ds x \in y\)

\(\implies\)

\(\ds x^+ \in y^+\)

Subset is Compatible with Ordinal Successor

\(\ds x \in y\)

\(\impliedby\)

\(\ds x^+ \in y^+\)

Sufficient Condition

\(\ds \leadsto \ \ \)

\(\ds x \in y\)

\(\iff\)

\(\ds x^+ \in y^+\)

$\blacksquare$