$(A=A\cup B )\Leftrightarrow B\subset A$ - is it true? I'm trying to find out whether the following statement is true:
$(A=A\cup B )\Leftrightarrow B\subset A$
In my opinion it is, because $A=A\cup B$ if B is either an empty set, or $A=B$ or $B \subset A$, all of which are are equivalent with $B \subset A$.
Could you please verify my proof?
Thanks

If $B \subset A$, then $A\cup B = A$. Indeed, $A \subset A\cup B \subset A\cup A = A$. Reciprocally, suppose $A = A\cup B$ and let us prove that $B \subset A$. If this was not the case, then there would be an $x \in B$ such that $x \not\in A$. But because $x \in B$, we obviously have $x \in A\cup B$, so that it cannot be $A = A\cup B$ (since the left side does not have $x$ and the right side does). 
