Singleton Equality

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Theorems

Let $x$ and $y$ be sets.

Then:

$\set x \subseteq \set y \iff x = y$
$\set x = \set y \iff x = y$


Proof

\(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle \set x \subseteq \set y\)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \forall z:\) \(\displaystyle \paren {z \in \set x \implies z \in \set y}\) Definition of Subset
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \forall z:\) \(\displaystyle \paren {z = x \implies z = y}\) Definition of Singleton
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \) \(\displaystyle x = y\) Equality implies Substitution

$\Box$


Then:

\(\displaystyle x\) \(=\) \(\displaystyle y\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \set x\) \(=\) \(\displaystyle \set y\) Substitutivity of Equality
\(\displaystyle \leadsto \ \ \) \(\displaystyle \set x\) \(\subseteq\) \(\displaystyle \set y\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle y\) by the first part

$\blacksquare$


Sources