Set Difference is Anticommutative

Theorem
Set difference is an anticommutative operation:


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

Proof

 * Suppose $$S = T$$.

Then $$S - T = \varnothing = T - S$$ from Set Difference Self Null.


 * Now suppose $$S - T = T - S$$.

Suppose $$S - T \ne \varnothing$$.

Let $$x \in S - T$$.

As $$S - T \subseteq S$$ from Set Difference Subset, we have that $$x \in S$$.

But then as $$S - T = T - S$$, we have that $$x \in T - S$$.

From the definition set difference, $$x \in T - S \implies x \notin S$$.

From this contradiction we conclude that $$S - T \ne \varnothing$$.

From Subset Equivalences it follows that $$S \subseteq T$$

Similarly we show that $$T - S = \varnothing$$, and hence that $$T \subseteq S$$

So by definition of set equality it follows that $$S = T$$.