Set is Element of Successor
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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, $x \in x^+$.
$\blacksquare$