What does this proposition mean?

Question

$∀x ∈ P(\Bbb{N}), x \notin \{\} \Rightarrow ∃y ∈ x, ∀z ∈ x \mid y < z$ Where $P(x)$ is the power set.

I'm interpreting it as "in all subsets of the natural numbers, there exists a value smaller than all values in the subset (including the value itself)". Since $z$ may be equal to $y$, the statement $y < z$ false. Should it be $y \le z$ or am I misinterpreting the proposition?

This is from a first-year algebra review sheet.

Answer 1

There is one more problem. The correct formulation is $$\forall x\in \mathcal P(\mathbb N): x\ne\emptyset \Rightarrow \exists y\in x\ \forall z\in x: y\le z$$ Note the $x\ne \emptyset$ vs. $x\notin\emptyset$ and $y\le z$ vs. $y<z$.

Answer 2

Yes, this is the Well-Ordering Principle: Every nonempty subset of the natural numbers contains a least element. The inequality should indeed be changed to be $y \leq z$.