Ordered Pair Formal Definition

The Kuratoiwski definition intends to enforce the one basic notion of an ordered pair, that is $$\langle a,b\rangle=\langle c,d\rangle\iff a=c\land b=d.$$ While one direction is trivial, note that $$\begin{align}&\langle a,b\rangle=\langle c,d\rangle\\ \implies&\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}\\ \implies&\{a\}\in\{\{c\},\{c,d\}\}\\ \implies&\{a\}=\{c\}\lor\{a\}=\{c,d\}\\ \implies&a=c\lor a=c=d\\ \implies&a=c\\ \end{align}$$ and then $$\begin{align}&\langle a,b\rangle=\langle a,d\rangle\\ \implies&\{\{a\},\{a,b\}\}=\{\{a\},\{a,d\}\}\\ \implies&\{a,b\}\in\{\{a\},\{a,d\}\}\\ \implies&\{a,b\}=\{a\}\lor \{a,b\}=\{a,d\}\\ \implies& b\in\{a\}\lor b\in\{a,d\}\\ \implies & b=a\lor b=d \end{align}$$ and by symmetry also $d=a\lor d=b$. Combined, this yields $(b=a\land d=a)\lor b=d$, i.e. $b=d$. In summary, $$\langle a,b\rangle=\langle c,d\rangle\implies a=c\land b=d.$$

Notice: $\langle a,b \rangle = \big\{\{a\},\{a,b\}\big\}$, but $\langle b,a \rangle = \big\{\{b\},\{a,b\}\big\}$. The first elements of these sets are different.