This statement should be true right? If $B \subseteq A$ and $x \in A$, $\exists x(x \in B)$. I was trying to prove this statement but my previous approach was wrong.
My previous approach involved misstating the property if $B \subseteq A$ then $B \land A = A$. This obviously is wrong as it should be $B \land A = B$.
So this is my new proof.
Given $B \subseteq A$, $B \lor A = A$. This means:
$\forall x(x \in (B \lor A) \leftrightarrow x \in A)$
The statement of interest is:
$\forall x(x \in A \rightarrow x \in (B \lor A)$
Thus if $x \in A$, x could be in B. Or that $\exists x(x \in B)$.