Prove that ≿ is transitive iff ≻ and ∼ are transitive (revamp)

Question

Above is my work so far. I know this question was asked before but it did not have a definite answer in it, which is why I'm asking it once more.

Answer 1

Suppose $\succ$ and $\sim$ are transitive and consider $x, y, z$ such that $x \succsim y$ and $y \succsim z$. Since $\succsim$ is a total relation, we must have $x \succsim z$ (in which case we're done), or $z \succsim x$. We will suppose the former is false, and hence the latter is true, in order to obtain a contradiction.

Therefore, $z \succ x$. If $\lnot(y \succsim x)$, then $x \succ y$, so by transitivity of $\succ$, we get $z \succ y$. This would contradict $y \succsim z$, hence $y \succsim x$, and thus $x \sim y$.

Similarly, if $\lnot(z \succsim y)$, then $y \succ z$. By the transitivity of $\succ$, we get $y \succ x$, contradicting $x \succsim y$. We similarly get $y \sim z$.

By transitivity of $\sim$, we see that $x \sim z \implies x \succsim z$ after all, which contradicts our assumption.