Definition:Smallest Set by Set Inclusion/Class Theory

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Let $A$ be a class.

Then a set $m$ is the smallest element of $A$ (with respect to the subset relation) if and only if:

$(1): \quad m \in A$
$(2): \quad \forall S: \paren {S \in A \implies m \subseteq S}$

Also known as

The smallest element in this context is also referred to as the least element.

Also see