Are these two statements regarding linear orders equivalent over ZF?

Question

I know that ZF set theory can't prove that every set can be linearly ordered. Now, consider the statement: "For every nonempty set $S$ and every element $s$ of $S$, $S$ can be linearly ordered with $s$ as the top element of that linear order". Is that statement equivalent, over ZF, to the statement that every set can be linearly ordered?

Answer 1

If you have a linear order $<$ on a set $S$ and if $s \in S$, you can perturb $<$ to make $s$ the top element. So the two statements are equivalent.