Singleton Equality

Theorems
Let $x$ and $y$ be sets.

Then:


 * $\left\{{x}\right\} \subseteq \left\{{y}\right\} \iff x = y$
 * $\left\{{x}\right\} = \left\{{y}\right\} \iff x = y$

Proof
The chained biconditionals in the first part satisfy the first part of the theorem.

Then:


 * $x = y \implies \{ x \} = \{ y \}$ (Substitutivity of equality)


 * $\{ x \} = \{ y \} \implies \{ x \} \subseteq \{ y \}$ (Theorem???)


 * $\{ x \} = \{ y \} \implies x = y$


 * $\{ x \} = \{ y \} \iff x = y$

Source

 * : $\S 7.7$