zorn's lemma and ascending chain condition

I am trying to figure out if the proof of the following well known fact
uses Zorn's lemma:

Let V be a partially ordered set. Suppose V satisifies the ascending
chain condition (every strictly increasing sequence x1<x2<x3< is
finite). Show that every non-empty subset of V has a maximal element.

The usual proof goes as follows: assume there is a nonempty subset T
that has no maximal element, then there exists element x1 in T since T
is non empty, and then there exists bigger element x2 in T since x1 is
not maximal, and then an x3 in T such that x1<x2<x3, and continue in
this manner. We get an infinite strictly increasing sequence, a
contradiction.

Am I using Zorn's lemma / axiom of choice here? It seems that in
constructing the sequence I'm making infinitely many arbitrary choices,
but I'm not sure.

Thanks,
Mike

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