Subset Relation is Antisymmetric

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Theorem

The subset relation is antisymmetric:

$\paren {x \subseteq y} \land \paren {y \subseteq x} \iff x = y$

where $x$ and $y$ are sets.


Proof

This is a direct statement of the definition of set equality:

$x = y := \paren {x \subseteq y} \land \paren {y \subseteq x}$

$\blacksquare$


Sources