Proving $A \subset B \implies A \cup B = B$ I have to prove this:
$A \subset B \implies A \cup B = B$
One direction:
$x\in A \subset B$
$\implies x \in A \land x \in B$
$\implies x \in A \cup B$ 
$\implies x \in (A \cup B = B)$
Other direction:
$x \in (A \cup B = B)$
$\implies x \in A \land x \in B$  (I used $\land$ here instead of $\lor$ because it equals B, so it has to be in both sets.)
$\implies x \in A \subset B$ 
Is my proof correct or are there some steps missing.
If not I hope you could show me the right direction.

Thanks for the help. I hope this is better.
Let $x \in A \subset B$.
$\Rightarrow x \in A \land x \in B \Rightarrow x \in (A \cup B) \Rightarrow x \in B \Rightarrow x \in (A \cup B = B)$. 
Assume $A \cup B = B$. 
Let $x \in A \Rightarrow x \in (A \cup B) \Rightarrow x \in B \Rightarrow x \in (A \subset B)$.
Could you please show me any mistakes or ways to improve it.
