I'm strugglin to prove set theory problem.
Let's define $(|a, b, c|) = \{\{a\}, \{a, b\}, \{a, b, c\}\}$. Prove that when $a\neq b\neq c\neq a$ and $a'\neq b'\neq c'\neq a'$ then $(|a, b, c|)=(|a', b', c'|)$ defines that $a=a', b=b', c=c'$.
If we leave out the rule of member of the sets need to be different, could $(|a, b, c|)=(|a', b', c'|)$ be true even when $(a, b, c)=(a', b', c')$?.
I tried to prove this by showing three steps how firstly if $\{a\}=\{a'\}$ then $a=a$ must be true. Because pair $\{a, b\}=\{ a', b'\}$ and as stated before $a=a$ then must be $b=b'$. And because $\{a, b, c\}=\{ a', b', c'\}$ and $a=a', b=b'$ must be $c=c'$.
What goes wrong? Also is there any example set how $(|a, b, c|)=(|a', b', c'|)$ even if $(a, b, c) \neq (a', b', c')$?