Definition:Smallest Element/Class Theory

Definition
Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be an ordering.

Let $A$ be a subclass of the field of $\RR$.

An element $x \in A$ is the smallest element :


 * $\forall y \in A: x \preceq y$

That is, $x$ strictly precedes, or is equal to, every element of $A$.

The smallest element of $A$ is denoted $\min A$.

For an element to be the smallest element, all $y \in A$ must be comparable with $x$.