Cowen's Theorem/Lemma 4

Lemma for Cowen's Theorem
Let $g$ be a progressing mapping.

Let $x$ be a set.

Let $\powerset x$ denote the power set of $x$.

Let $M_x$ denote the intersection of the $x$-special subsets of $\powerset x$ $g$.

Let $M$ be the class of all $x$ such that $x \in M_x$.

We have that:
 * $M$ is closed under chain unions.

Proof
Let $C$ be a chain of elements of $M$.

For each $x \in C$, we have $x \subseteq \ds \bigcup C$.

Hence by Lemma $3$:
 * $M_x \subseteq M_{\mathop \cup C}$

Also, because $x \in M_x$, we have:
 * $x \in M_x$

Hence:
 * $x \in M_{\mathop \cup C}$

Thus:
 * $C \subseteq M_{\mathop \cup C}$

Because $M_{\mathop \cup C}$ is closed under chain unions:
 * $\ds \bigcup C \in M_{\mathop \cup C}$

Hence:
 * $\ds \bigcup C \in M$