The Law of the Excluded Middle (LEM) states that for any proposition $p$, we have $\vdash p \lor \lnot p $.
Syntactic completeness (a.k.a negation completeness) states that for any proposition $p$, we have $\vdash p$ or $\vdash \lnot p$.
As far as I'm aware, in classical propositional logic the former implies the latter (what's the simplest way to justify this?). This is highly problematic though, because it would mean that the contrapositive LEM is false) renders classical (Peano) arithmetic inconsistent – that is, LEM cannot possibly be a valid axiom/rule.
This strikes me as plain wrong, from what I've read. So, where have I messed up in my reasoning? Can we not in fact say that $\vdash p \lor \lnot p $ implies $\vdash p$ or $\vdash \lnot p $, at least not for classical logic? It seems intuitively true, but since I'm struggling to justify it formally, perhaps this is where the mistake lies.