Restricted Tukey's Theorem/Weak Form

Theorem
Let $X$ be a set.

Let $\mathcal A$ be a non-empty set of subsets of $X$.

Let $'$ be a unary operation on $X$.

Let $\mathcal A$ have finite character.

For all $A \in \mathcal A$ and all $x \in X$, let either:
 * $A \cup \set x \in \mathcal A$

or:
 * $A \cup \set {x'} \in \mathcal A$

Then there exists a $B \in \mathcal A$ such that for all $x \in X$, either $x \in B$ or $x' \in B$.