Proof for $A \cup B = B$ if and only if $A \subset B$
$$(x \in A) \implies (x\in A\cup B=B)\implies (x\in B)\implies (A\subset B).$$ Conversely, Let $A\subset B$. $$x\in A\cup B\implies (x\in B)\implies (A\cup B=B)$$
A U B implies x is in A or x exist in B . But since AUB=B , x exist in B .
That step is small but important in the explanation. I hope it helps
The fact that AUB=B is one of your givens. You use that statement to build your argument to show that ultimately A is a subset of B