Axiomatic System Number Two (3/18/14-Current)

Let $\_∘\_\{\_\}$ be a trinary relation, pronouncing $a∘b\{c\}$ as "$a$ is essential to $b$ in universe $c$," or "$a$ essential to $b$ in $c$." Accept the following axioms:

1. (Axiom of Equality) Define $a\text{~} b\{c\}:⇔∀z(z∘a\{c\}⇔z∘b\{c\})$ ($a$ similar to $b$ in $c$), which is seen to be an equivalence relation in any fixed universe $c$. Then the axiom is $∀a,b(∃x,y[x≠y∧a\text{~} b\{x,y\}]⇒a=b)$, i.e. two objects $a$ and $b$ are equal (where equality can be formally expanded as the property that any trinary relation containing an instance of $a$ can be rewritten with that $a$ replaced by $b$, and vice versa) if $a\text{~} b$ in two distinct universes $x,y$.
2. (Axiom of Iteration) $∀x∃\overline{x}∀a,b(a∘\overline{x}\{b\}⇔[a∘x\{b\}∨a=x])$, i.e. for every $x$ there exists an iteration $\overline{x}$ such that $\overline{x}$'s essentials are the essentials of $x$ in that particular universe plus the additional essential of $x$ itself (if it were not already essential).
3. (Axiom of Existence) $∃∅∀a,b(¬[a∘∅\{b\}])$. Combining the axioms of iteration and existence, we see that for "any" (uniqueness not yet established) $∅$, we have $\overline{∅}≠∅$ as $∅∘\overline{∅}\{∅\}$ yet $¬(∅∘∅\{∅\})$, so there are in fact at least two distinct objects. Now statements defining explicitly the essentials of an object in any given universe (like the existence of $∅$ and $\overline{x}$) can be taken as unique existences.
4. (Axiom of Tensors) $∀x,y∃z(z\text{~} x\{y\}∧z\text{~} y\{x\})$, i.e. there exists an object imitating $x$ in $y$ and $y$ in $x$. When $x≠y$, we see this $z$ is unique by the axiom of equality, so we define $x⊗y$ to be this unique $z$ when $x≠y$ and we define $x⊗x=x$. Note $⊗$ is commutative.

Lemma 1 (Description of Trivial Tensors): $x⊗y=x⇔x\text{~} y\{x\}$. Proof: ($⇒$) We observe $x=(x⊗y)\text{~} y\{x\}$. ($⇐$) We observe $(x⊗y)\text{~} x\{y\}$ and $(x⊗y)\text{~} y\text{~} x\{x\}$, so when $x≠y$ we have $x⊗y=x$; when $x=y$ we definitionally observe $x⊗x=x$.$~\square$

Lemma 2 (Description of Tensor Injectivity): When $y≠z$, we have $x⊗y=x⊗z⇔x⊗y=x⊗z=x⇔x\text{~} y\text{~} z\{x\}$. Proof: The second logical equivalence is an application of the first lemma, so we prove now the first equivalence: ($⇐$) This is trivial. ($⇒$) We have $(x⊗y)\text{~} x\{y\}$ and also $(x⊗z)\text{~} x\{z\}$. Since these two tensors are equivalent, when $y≠z$ we observe this tensor must be equivalent to $x$.$~\square$ This lemma generalizes the first, and interpreted conceptually says that $x⊗\_$ is "injective" on the collection of objects not similar to $x$ in its own universe (with e.g. $x$ appended to this collection). Unfortunately, it is not enough to know the fixed similarity class of an object $y$ defined explicitly in the universe of $x$, if wanting to uniquely determine $x⊗y$.

Proposition 3 (Nonassociativity of $⊗$): Generally, we do not have $x⊗(y⊗z)=(x⊗y)⊗z$. Proof: We show particularly that $∅⊗(∅⊗\overline{∅})≠(∅⊗∅)⊗\overline{∅}=∅⊗\overline{∅}$. Assume the contrary and apply lemma 2 (after showing $∅⊗\overline{∅}≠\overline{∅}$ by appealing to lemma 1) to particularly obtain $∅⊗\overline{∅}=∅$. Apply lemma 1 to obtain $\overline{∅}\text{~} ∅\{∅\}$, even though $¬(∅∘∅\{∅\})$ but $∅∘\overline{∅}\{∅\}$.$~\square$