Exercise 2, § 2

(a) Show that the relation $\{\{x\}, \{x, y\}\} = \{\{x'\}, \{x', y'\}\}$ is equivalent to "$x = x'$ and $y = y'$".

(b) Let $\mathscr{T}_0$ be the theory of sets and let $\mathscr{T}_1$ be the theory which has the same schemes and explicit axioms as $\mathscr{T}_0$, except for axiom A3. Show that if $\mathscr{T}_1$ is not contradictory, then $\mathscr{T}_0$ is not contradictory (use (a)).