Axiom of Swelledness is implied by Axiom of Replacement

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

Then the Axiom of Swelledness holds.

Proof
Recall the Axiom of Replacement:

Recall the Axiom of Swelledness:

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 Axiom of the Empty Set, $A$ 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 Axiom of Replacement:
 * $A$ is therefore also a set.

Hence the result.