Empty Intersection iff Subset of Complement/Proof 2

Corollary to Intersection with Complement is Empty iff Subset

$S \cap T = \O \iff S \subseteq \relcomp {} T$

$S \subseteq T \iff S \cap \relcomp {} T = \O$

Then we have:

$\ds $

$\ds S \nsubseteq \relcomp {} T$

$\ds $

$\leadstoandfrom$

$\ds \neg \paren {\forall x \in S: x \in \relcomp {} T}$

Definition of Subset

$\ds $

$\leadstoandfrom$

$\ds \exists x \in S: x \notin \relcomp {} T$

$\ds $

$\leadstoandfrom$

$\ds \exists x \in S: x \in T$

Definition of Set Complement

$\ds $

$\leadstoandfrom$

$\ds x \in S \cap T$

$\ds $

$\leadstoandfrom$

$\ds S \cap T \ne \O$

Definition of Disjoint Sets

Thus:

$\ds $

$\ds S \cap T = \O$

$\ds $

$\leadstoandfrom$

$\ds \forall x \in S: x \in \relcomp {} T$

$\ds $

$\leadstoandfrom$

$\ds S \subseteq \relcomp {} T$

$\blacksquare$