Definition:Supercomplete Class

Definition

Let $A$ denote a class.

Then $A$ is a supercomplete class if and only if both:

$A$ is transitive

and:

$A$ is swelled.

That is, $A$ is supercomplete if and only if:

$\forall x: \forall y: \paren {x \in y \land y \in A \implies x \in A}$

$\forall x: \forall y: \paren {x \subseteq y \land y \in A \implies x \in A}$

Also see

• Definition:Transitive Class
• Definition:Swelled Class

• Results about supercomplete classes can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness