Show that $T$ is inconsistent

Question

Let $L$ be the language $\{\cdot, e\}$ and let $T$ be the $L$-theory whose proper axioms are:

1. $(\forall x)(\forall y)(\forall z)((x.y).z=x.(y.z)) $
2. $(\forall x)(x.e=x \,\wedge \, e.x=x)$
3. $(\forall x)(\exists y)(x.y=e \,\wedge \, y.x=e)$
4. $(\forall x)((x.(x.(x.x)))=e)$
5. $(\exists x)(\neg (x=e) \, \wedge \, x.(x.x)=e )$

Show that $T$ is inconsistent

My attempt:

From $5$th axiom we know $x.x.x=e$, so by $4$th axiom we get $x.e=e$ . But according to $2$nd axiom $x$ must be equal to $e$.

So, $T$ is inconsistent because $5$th axiom ,which says $(\exists x)(\neg (x=e))$, doesn't hold now, since $x=e$ . But I somehow doubt that this isn't a correct way of showing T is inconsistent. How can I improve the solution?

Answer 1

Your approach is correct; at most it needs rewording for clarity, in case a reader forgets axiom 5's existence claim applies to the same $x$ each time you invoke it. (I actually thought you'd made a mistake at first, but that's only because correct proofs can be hard to read.) Fixing an $x$ that's an example of 5, $e=x^4=ex=x$, a contradiction. As you've noted, the $=$ signs respectively use 4, 5, 2; then 5 gives the contradiction.