User:Dfeuer/Equality of Ordered Pairs implies Equality of Elements/Lemma 1

Theorem

Let $a$, $b$, and $c$ be sets.

Then:

$\{a\} = \{b, c\} \iff a = b = c$

Proof

Reverse implication

Suppose that $a = b = c$.

Let $x \in \{a\}$.

Then by the definition of User:Dfeuer/Definition:Singleton, $x = a$.

Since $a = b$, $x = b$.

Thus by the definition of User:Dfeuer/Definition:Unordered Pair, $x \in \{b,c\}$.

Let $x \in \{b, c\}$.

By the definition of unordered pair:

$x = b$ or $x = c$.

Since $a = b = c$, $x = a$.

Thus by the definition of singleton, $x \in \{a\}$.

We have shown that $\forall x: (x \in \{a\} \iff x \in \{b, c\}$.

By the User:Dfeuer/Axiom of Extensionality, $ \{a\} = \{b, c\}$

$\Box$

Reverse implication

Suppose that $\{a\} = \{b, c\}$.

By the definition of unordered pair, $b \in \{b,c\}$ and $c \in \{b,c\}$.

Thus $b \in \{a\}$ and $c \in \{a\}$.

By the definition of singleton: $a = b$ and $a = c$.

$\blacksquare$