# Singleton Equality

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## 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

\(\displaystyle \left\{ {x}\right\} \subseteq \left\{ {y}\right\}\) | \(\iff\) | \(\displaystyle \forall z: \left({z \in \left\{ {x}\right\} \implies z \in \left\{ {y}\right\} }\right)\) | Definition of Subset | ||||||||||

\(\displaystyle \) | \(\iff\) | \(\displaystyle \forall z: \left({z = x \implies z = y}\right)\) | Definition of Singleton | ||||||||||

\(\displaystyle \) | \(\iff\) | \(\displaystyle x = y\) | Equality implies Substitution |

$\Box$

Then:

\(\displaystyle x = y\) | \(\implies\) | \(\displaystyle \left\{ {x}\right\} = \left\{ {y}\right\}\) | Substitutivity of Equality | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \left\{ {x}\right\} \subseteq \left\{ {y}\right\}\) | |||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle x = y\) | by the first part |

$\blacksquare$

## Sources

- 1963: Willard Van Orman Quine:
*Set Theory and Its Logic*: $\S 7.7$