# Set is Element of Successor

## Theorem

Let $x$ be a set.

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

Then $x \in x^+$.

## Proof

By the definition of successor set:

$x^+ = x \cup \{x\}$.

By the definition of singleton, $x \in \{x\}$.

Thus by the definition of union, $x \in x^+$.

$\blacksquare$