Prove { {a}, {a,b} } = { {c}, {c,d} } if and only if a=b and c=d.
I'm wondering how to proceed. I'm not even sure what exactly am I trying to prove
For sets A,B the definition tells us that $\displaystyle A=B\Longleftrightarrow A\subset B\,\,\,and\,\,\,B\subset A$ , so:
$\displaystyle \{ \{a\}, \{a,b\} \} = \{ \{c\}, \{c,d\} \} \Longrightarrow \{ \{a\}, \{a,b\} \} \subset \{ \{c\}, \{c,d\} \} \Longrightarrow \{a\}\in \{ \{c\}, \{c,d\} \} $ , and so $\displaystyle \{a\}=\{c\}\,\,\,or\,\,\,\{a\}=\{c,d\}$ .
But $\displaystyle \{a\}$ contains one single element so this set cannot be equal to a set containing two elements, and thus $\displaystyle \{a\}=\{c\}\Longleftrightarrow a=c$ . Take it from here.
Tonio