"I went on the internet today", as it were, and happened upon a devious proof for the claim that either a thing $X$ exists, or there exists a proof that $X$ does not exist.
The proof, if I understood its author correctly, is as follows:
Define proof to mean a valid argument with all true premises. Let $P_1$ be the proposition that $X$ exists. Then
Either $X$ exists, or $X$ does not exist: $P_1 \vee \neg P_1$ is true by assumption. If $X$ exists, then our claim is satisfied and we may stop here; otherwise:
Let us assume that there does not exist a proof that $X$ does not exist.
Disjunctive syllogism, $((P \vee Q) \wedge \neg P) \rightarrow Q $, is a valid argument, with premises $P \vee Q$ and $\neg P$.
$((P_1 \vee \neg P_1) \wedge \neg P_1) \rightarrow \neg P_1 $ is a disjunctive syllogism, thus a valid argument, with premises $P_1 \vee \neg P_1$ and $\neg P_1$.
If $((P_1 \vee \neg P_1) \wedge \neg P_1) \rightarrow \neg P_1 $ has all true premises, it is a proof (since it is a valid argument). But this violates our assumption in step 2, so at least one of the premises $P_1 \vee \neg P_1$ and $\neg P_1$ must be false.
$P_1 \vee \neg P_1$ is true by assumption, so of the two premises only $\neg P_1$ can be false.
Therefore, $\neg P_1$ is false, thus $P_1$ is true: if there is no proof that $X$ does not exist, then $X$ has been shown to exist!
...now this feels extremely wrong, but I simply cannot tell why. I'm almost certain that there is an error at least in step 5, but I cannot see it. I can't even tell if the proof contains a logical error or if it has employed some form of verbal trickery. The proof seems correct, yet feels wrong.
What error(s), if any, have been made in the above proof? Is the original claim correct even if the proof itself fails?