In Gödel's Incompleteness Theorems book, Smullyan presents a proof of the Separation Lemma for sets. Let be S a system, where all formulas of $\Omega_4$ and $\Omega_5$ (axiom schemes of (R)) are provable in S. Let A and B any sets. He wants to show that B-A is separable from A-B by the formula: $$\forall y(A(x,y)\supset (\exists z\leq y) B(x,z))$$ where: the formula A(x,y) enumerates the set A, and the formula B(x,y) enumerates the set B.
Part of the proof (and my question):
- Suppose $n\in B-A$. Then $n\in B.$ So, for some k, the sentence $B(\overline n, \overline k)$ is provable.
Also $n \notin A$, and so for for every $m\leq k$, the sentence $A(\overline n, \overline m)$ is refutable, and, by $\Omega_4$, the sentence $(\forall y\leq \overline k)\sim A(\overline n, y)$ and the open formula $y\leq \overline k \supset \sim A(\overline n, y)$ are provable.
By using $\Omega_5$, $ A(\overline n, y) \supset \overline k\leq y$ is provable.
Then, since $B(\overline n, \overline k)$ is provable, then $A(\overline n, y) \supset ((\overline k\leq y)\land B(\overline n, \overline k))$ is provable.
My question is about this last statement.
I think if $A\supset B$ and C are provable, it's not correct to deduce that $A\supset (B \land C)$ is provable.
Thanks.