User:Dfeuer/Singletons are Equal iff Elements are Equal
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Theorem
Let $a$ and $b$ be sets.
Let $\{a\}$ and $\{b\}$ be User:Dfeuer/Definition:Singleton $a$ and singleton $b$, respectively.
Then:
- $\{ a \} = \{ b \} \iff a = b$
Proof
Suppose that $\{ a \} = \{ b \}$.
By the definition of User:Dfeuer/Definition:Singleton:
- $\forall x: x \in \{a\} \iff x = a$
Thus $a \in \{a\}$.
Since $\{ a \} = \{ b \}$, $a \in \{ b \}$.
By the definition of singleton:
- $\forall x: x \in \{b\} \iff x = b$
Thus $a = b$.
Suppose instead that $a = b$.
By the definition of singleton:
- $\forall x: x \in \{a\} \iff x = a$
- $\forall x: x \in \{b\} \iff x = b$
Since $a = b$:
- $\forall x: x \in \{b\} \iff x = a$
Thus:
- $\forall x: x \in \{b\} \iff x \in \{a\}$
By the User:Dfeuer/Axiom of Extensionality, $\{a\} = \{b\}$.
$\blacksquare$