Proving set theory subsets using element argument How do you even prove a set theory subset statement using element argument? I simply just can't find any relevance to the question with the notes i was studying.
Any guidance would be much appreciated.

(A - B) ∩ (C - B) subset of (A ∩ C) - B

The only definition i have is 

x element of A - B is logically equivalent to ( x element of A and x
  not element of B)

If i were to sub in the definition, it would lead me to nowhere where i can use whatever law there is in set theory.
This discrete mathematics is way different from the typical maths i have been doing since young. Any guidance is appreciated. 
 A: Note that 
$$x\in(A-B)\cap (C-B) $$
is equivalent to
$$x\in A-B\quad \land \quad x\in C-B $$
and so to
$$(x\in A \land x\notin B)\land (x\in C\land x\notin B). $$
Form this you want to show that $x\in(A\cap C)-B$, or equivalently, that
$$(x\in A\land x\in C)\land x\notin B. $$
I guess you can take it from here.
A: You can do this simply by using strict definitions of subsets:
$A \subset B$ is saying $\forall x\in A$ $x\in B$
So:
Let $x\in (A-B)\cap(C-B)$
Then $x\in (A-B)$ and $x\in(C-B)$
Then $x\in A\cap C$ and $x\notin (A\cap C)\cap B$ otherwise contradiction to the assumption $x\in (A-B)\cap(C-B)$
Then $x\in (A\cap C)-B$
A: As part of gaining deeper insights I'm trying to answer questions so that errors in my reasoning will hopefully be pointed out.
To proof using the element argument:
${(A-B)\cup(C-B) \subseteq (A \cup C)-B}$
We start as follows
${x \in (A-B)\cap(C-B)}$
${x \in (A \cap \overline{B})\cap(C\cap\overline{B}) }$ difference law
${(x \in A \land x \in \overline{B}) \land (x \in C \land x \in \overline{B})}$ def union
${(x \in A \land x \in C) \land (x \in \overline{B} \land x \in \overline{B})}$ associative + commutative laws
${(x \in A \land x \in C) \land x \in \overline{B}}$ idempotency law
${x \in (A \cap C) \cap \overline{B}}$ def intersection
${x \in (A \cap C)-B}$ difference law
And that concludes the proof.
This is the long version of what Hagen von Eitzen has given.
A: [ (A - B) ∩ (C - B) ] ⊆ [ (A ∩ C) - B ]
steps:
1.
∈(−)∩(C−)
2.
(a) ∈(−) and
(b) ∈(C−)
3.
From (a) ∈ and x∉B
From (b) ∈C and x∉B
4.
Hence, ∈ and ∈C and x∉B are true.
5.
By definition of intersection: ∈(∩C) and x∉B
6.
By definition of set difference: ∈(∩C)-B
