Union is Empty iff Sets are Empty

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 2
Let $S \cup T = \varnothing$.

We have:

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

So it follows by Equality of Sets that $S = \varnothing$.

Similarly for $T$.