basic proof regarding subsets

Question

If you want to prove $A \cap B$ $\subseteq$ $A\cup B\ $

then if you suppose

$x\in A \cap B$

$x\in A$ and $x\in B$

$x\in A$

$x\in A$ or $x\in B$

$x\in$ $A\cup B\ $

So what i do not understand here is the need for line 3 $x\in A$. If you took this line out would the proof be valid by definition of logical or? If not why is this line necessarys. Thank you.

Answer 1

Since line $2$ includes line $3$, there is no need for line $3$ in this proof.

However including line $3$ does not make the proof wrong.

It somehow legitimize line $4$ more clearly than line $2$ alone.

Answer 2

Line 3 can be safely taken out, since $a\land b\implies a\lor b$ is a logical (not set-theoretical) truth. Here $a=x\in A$ and $b=x\in B$.