Relative Complement of Decreasing Sequence of Sets is Increasing

Theorem
Let $X$ be a set.

Let $\sequence {E_n}_{n \mathop \in \N}$ be a decreasing sequence of sets in $X$.

Then $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an increasing sequence of sets in $X$.

Proof
Since $\sequence {E_n}_{n \mathop \in \N}$ is decreasing, we have:


 * $E_{n + 1} \subseteq E_n$ for each $n \in \N$.

From Relative Complement inverts Subsets, we then have:


 * $X \setminus E_n \subseteq X \setminus E_{n + 1}$ for each $n \in \N$.

So $\sequence {X \setminus E_n}_{n \mathop \in \N}$ is an increasing sequence of sets in $X$.