Boolean Prime Ideal Theorem

Theorem
Let $\struct {S, \le}$ be a Boolean algebra.

Let $I$ be an ideal in $S$.

Let $F$ be a filter on $S$.

Let $I \cap F = \O$.

Then there exists a prime ideal $P$ in $S$ such that:
 * $I \subseteq P$

and:
 * $P \cap F = \O$

Motivation
The Boolean Prime Ideal Theorem is weaker than the Axiom of Choice, but is similarly independent of ZF theory.

It is sufficient to prove a number of important theorems, although such proofs are often more involved than ones relying on the Axiom of Choice.