De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union

Theorem
Let $S$ and $T$ be sets.

Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.

Then:
 * $\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$

where:
 * $\displaystyle \bigcap_{i \mathop \in I} T_i := \set {x: \forall i \in I: x \in T_i}$

that is, the intersection of $\family {T_i}_{i \mathop \in I}$.

Proof
Suppose:
 * $\displaystyle x \in S \setminus \bigcup_{i \mathop \in I} T_i$

Note that by Set Difference is Subset we have that $x \in S$ (we need this later).

Then:

Therefore:
 * $\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$