De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection/Proof

Proof
Suppose:
 * $\displaystyle x \in S \setminus \bigcap \mathbb T$

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

Then:

Therefore:
 * $S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$