Successor in Limit Ordinal

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Theorem

Suppose that $x$ is a limit ordinal and that $y \in x$.


Then $y^+ \in x$ where $y^+$ denotes the successor set of $y$:

$\forall y \in x: y^+ \in x$


Proof

Since $x$ is a limit ordinal:

$x \ne y^+$

Moreover by Successor of Element of Ordinal is Subset:

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


Therefore by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:

$y^+ \subset x$ and $y^+ \in x$

$\blacksquare$