Definition:Smallest Set by Set Inclusion/Class Theory

Definition

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

• Definition:Greatest Set by Set Inclusion (Class Theory)

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.8$