Axiomatic System Number One (Inconsistent) (3/17/14)

Let $·$ be a binary relation, and accept the following axioms:

1. (Axiom of Existence) $∃X∀y(y·X⇔y=X)$, i.e. there exists "existential $X$" such that it itself is the one and only thing essential to it.
2. (Axiom of Equality) $∀x,y(x=y⇔∀z[z·x⇔z·y])$, where here $x=y:⇔∀z([x·z⇔y·z]∧[z·x⇔z·y])$. This should be familiar from set theory.
3. (Axiom of Transitivity) $∀a,b,c(a·b·c⇒a·c)$, i.e. $·$ is transitive.
4. (Axiom of Antitheses) $∀x∃A(x)∀y(y·A(x)⇔¬[y·x])$, i.e. $A(x)$ flips all the essential relations toward $x$. $A(x)$ is unique according to the second axiom.
5. (Axiom of Agreement) $∀x∃y(x,A(x)·y)$. This $y$ (called an agreement of $x$) may not be unique for any given $x$, and by $x'$ we denote any particular agreement that remains fixed throughout the proof in question.

This system is inconsistent.

Proof: We see $X≠A(X)$ as $X·X$ yet $¬(X·A(X))$ by the definition of $A(X)$. As well, $X≠X'$ seeing as $A(X)·X'$ yet $¬(A(X)·X)$ since we just proved $X≠A(X)$. Therefore $¬(X'·X)$, hence we have $X·X'·A(X)$ implying $X·A(X)$, a contradiction.$~\square$

This system wasn't getting interesting enough to make it worth repairing. Originally there was a second binary relation, but there wasn't enough interplay between the two relations, and the second was just generally neglected, hence it didn't appear in the proof of inconsistency so its axioms weren't included.