Set Difference Equals First Set iff Empty Intersection

Theorem

 * $S - T = S \iff S \cap T = \varnothing$

Proof
Suppose $S - T = S$. Then, by definition of set equality, we have $S \subseteq S - T$. Thus:

Thus we have that $S - T = S \implies S \cap T = \varnothing$.

Similarly, we have $S \cap T = \varnothing \implies S \subseteq S - T$ from the above.

But from Set Difference Subset we also have $S - T \subseteq S$.

Thus by the definition of set equality, it follows that $S \cap T = \varnothing \implies S - T = S$.

The implications therefore go both ways and the proof is complete.