Principle E is Equivalent to Kuratowski's Lemma

Theorem
Kuratowski's Lemma is equivalent to Principle E.

Proof
Recall Kuratowski's Lemma:

Recall Principle E:

Kuratowski's Lemma implies Principle E
Let Kuratowski's Lemma hold.

Let $x$ be a maximal element under the subset relation.

Then $x$ has no immediate extension which is not $x$ itself.

That is, Principle E holds directly.

Principle E implies Kuratowski's Lemma
Let Principle E hold.

Let $S$ be swelled.

That is, every subset of every element of $S$ is an element of $S$.

Then an element $x$ of $S$ has no immediate extension in $S$ $x$ is a maximal element of $S$ under the subset relation.

Thus Principle E implies that for every swelled set $S$ which is closed under chain unions, every $b \in S$ is a subset of some maximal element of $S$.

By Class of Finite Character is Swelled, a set of finite character is swelled.

By Class of Finite Character is Closed under Chain Unions a set of finite character is closed under chain unions.

Thus Principle E implies Tukey's Lemma.

From Maximal Principles are Equivalent, we have that Tukey's Lemma is equivalent to Kuratowski's Lemma.