Prove or disprove $(A-B)\cup(B-C)=A-C$

Question

Prove or disprove the following statement: $(A-B)\cup(B-C)=A-C$

I know is true, but can I use a typical element argument to prove it?
I need to split into 2 parts

1. $(A-B)\cup (B-C)\subseteq A-C$

2. $A-C\subseteq (A-B)\cup (B-C)$

then I am not sure how to continue.
Is it start by $x\in (A-B)\cup (B-C)\implies x\in(A\cap \overline{B})\cup (B\cap\overline{C})$ ?

Answer 1

The claim is false. Take for example $B=\emptyset$ and $A=C\not=\emptyset$ then $$(A-B)\cup (B-C)=A\quad \text{and}\quad A-C=\emptyset.$$