Union equals Intersection iff Sets are Equal

Theorem
Let $$S$$ and $$T$$ be sets.

Then:
 * $$S \cup T = S, S \cap T = S \iff S = T$$

where:
 * $$S \cup T$$ denotes set union;
 * $$S \cap T$$ denotes set intersection.

Proof
From Intersection with Subset is Subset, we have:
 * $$S \subseteq T \iff S \cap T = S$$

From Union with Superset is Superset, we have:
 * $$S \subseteq T \iff S \cup T = T$$

That is:


 * $$T \subseteq S \iff S \cup T = S$$

Thus:
 * $$S \cup T = S, S \cap T \iff S \subseteq T \subseteq S$$

From Equality of Sets, we have:
 * $$S = T \iff S \subseteq T \subseteq S$$

Hence the result.