# Empty Intersection iff Subset of Complement/Proof 2

## Corollary to Intersection with Complement is Empty iff Subset

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

## Proof

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

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 $\ds$ $\leadstoandfrom$ $\ds \exists x \in S: x \notin \relcomp {} T$ Denial of Universality $\ds$ $\leadstoandfrom$ $\ds \exists x \in S: x \in T$ Definition of Set Complement $\ds$ $\leadstoandfrom$ $\ds x \in S \cap T$ Definition of Set Intersection $\ds$ $\leadstoandfrom$ $\ds S \cap T \ne \O$ Definition of Disjoint Sets

Thus:

 $\ds$  $\ds S \cap T = \O$ $\ds$ $\leadstoandfrom$ $\ds \forall x \in S: x \in \relcomp {} T$ $\ds$ $\leadstoandfrom$ $\ds S \subseteq \relcomp {} T$

$\blacksquare$