$(A=A\cup B )\Leftrightarrow B\subset A$ - 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

$$B\not\subseteq A\iff \exists x\in B\setminus A\iff x\in A\cup B, x\notin A \iff A\neq A\cup B$$
