Yesterday I pose a question where I ask about the correctness of a proof where I shown that the Axiom of Choice implies that for any not empty collection of not empty sets the projection defined on their cartesian product are surjective funcions, so then I asked to me if from the assertion about the surjectivity of the projections I can argue the Axiom of Choice. Here following the link of the question where I proved the first implication: link.
Well could someone help me, please?
The Axiom of Choice is that if $\{S_i\}_{i\in I}$ is a non-empty indexed family of non-empty sets (...which means there is a function $g$ with domain $I\ne \phi,$ and $g(i)=S_i\ne \phi$ for each $i\in I$...) then there exists a Choice function $f:I\to\bigcup \{S_i\}_{i\in I},$ that is, $f(i)\in S_i$ for each $i\in I.$ In other words there exists $f\in\prod_{i\in I}S_i,$ that is, $\prod_{i\in I}S_i\ne \phi.$
If $I\ne \phi$ and each $S_i\ne \phi$ and if a projection $p_j:\prod_{i\in I}S_i\to S_j$ is surjective for any $j\in I$ then $\prod_{i\in I}S_i\ne\phi.$