Axiom of Swelledness is implied by Axiom of Replacement

Theorem
Let the (in the context of class theory) be accepted.

Then the holds.

Proof
Recall the Axiom of Replacement:

Recall the :

That is:
 * Every subclass of a set is a set.

Let $x$ be a set.

Let $A$ be a class such that $A \subseteq x$.

Suppose $A$ is the empty class.

Then by the, $A$ is a set.

Suppose $A$ is a non-empty class.

Let $c \in A$ be arbitrary.

Let $f$ be:
 * the class of all ordered pairs $\tuple {a, a}$ for $a \in A$

along with:
 * all ordered pairs $\tuple {y, c}$ where $y \in x \setminus A$.

Thus:
 * $f$ is a mapping whose domain is $x$

and:
 * $\forall y \in x: \map f y = \begin {cases} y & : y \in A \\ c & : y \notin A \end {cases}$

Thus:
 * $\forall y \in x: \map f y \in A$

and in fact:
 * $f \sqbrk x = A$

We have that $x$ is a set.

Then by the :
 * $A$ is therefore also a set.

Hence the result.