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