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

Statement

For every set $A$ and property $P$ there is a unique set $B$ such that $x \in B$ iff $x \in A$ and $P(x)$. This set is denoted $\{x \in A \mid P(x)\}$.

Explanation

Proof(s)

By the Axiom Schema of Comprehension we have that there exists a set $B$ such that $x \in B$ iff $x \in A$ and $P(x)$.
Suppose there exist two such sets, $B$ and $B'$.
Notice that by their definitions, $B$ and $B'$ contain precisely the same elements.
Hence by the Axiom of Extensionality we have that $B = B'$, as wanted.

History

Applications

Links

Dependencies

• Axiom of Extensionality
• Axiom Schema of Comprehension

Dependents

Sources

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