Prove or disprove the axiom of pairing impllies the weak axiom of pairing.
Solution 1:
I think that you just have a small misunderstanding when interpreting the statement---in symbols, the intended meaning is: $$ x \in C \iff (x = A \vee x = B). $$ In particular, if $x = A$ then certainly $x = A \vee x = B$, and thus $x \in C$ (i.e. $A \in C$). Likewise for $B$, so no, the axiom of pairing "does always imply that $A \in C$ and $B \in C$".
I think your interpretation has followed from misinterpreting the "$\iff$" as an "$\implies$".