Union is Empty iff Sets are Empty/Proof 2

Theorem
If the union of two sets is the empty set, then both are themselves empty:


 * $S \cup T = \varnothing \iff S = \varnothing \land T = \varnothing$

Proof
Let $S \cup T = \varnothing$.

We have:

Then we have $\varnothing \subseteq S$ from Empty Set Subset of All.

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

Similarly for $T$.