Uniqueness of Power Set - kipawaa/Proof-Tree GitHub Wiki

Statement

For any set $S$, there exists a unique set $P$ such that $X \in P$ iff $X \subseteq S$.

Explanation

Proof(s)

By the Axiom of Power Set there exists a set $P$ such that $X \in P$ iff $X \subseteq S$.
Suppose there exist two such sets, $P$ and $P'$.
Notice that $P$ and $P'$ contain exactly the subsets of $S$ and hence contain precisely the same elements.
Hence by the Axiom of Extensionality we have that $P = P'$, a wanted.

History

Applications

Links

Dependencies

• Axiom of Extensionality
• Axiom of Power Set

Dependents

Sources

Hrbacek, K., & Jech, T. (1999). Introduction to Set Theory, Revised and Expanded (3rd ed.). CRC Press. https://doi.org/10.1201/9781315274096