Set Union/Examples/Subset of Union

Example of Set Union

Let $U, V, W$ be non-empty sets.

Let $W$ be such that for all $w \in W$, either:

$w \in U$

or:

$w \in V$

Then:

$W \subseteq U \cup V$

Proof

 $\displaystyle w$ $\in$ $\displaystyle W$ $\displaystyle \leadsto \ \$ $\displaystyle w$ $\in$ $\displaystyle U$ by hypothesis $\, \displaystyle \lor \,$ $\displaystyle w$ $\in$ $\displaystyle V$ $\displaystyle \leadsto \ \$ $\displaystyle w$ $\in$ $\displaystyle U \cup V$ Definition of Set Union $\displaystyle \leadsto \ \$ $\displaystyle W$ $\subseteq$ $\displaystyle U \cup V$ Definition of Subset

$\blacksquare$