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.

Below a proof using the natural deduction method. 
The goal is read as a simple implication : if A is included in B , then ... ( not as a double implication). 
