In a prior posting, I've posed the idea of reducing set theory to an extended kind of second order ordinal arithmetic $\small \sf`` 2 oO A"$. The idea was to have a domain of ordinals and sets of ordinals where the set of all ordinals is well ordered by relation $<$ denoting ordinal strict smaller than, that have an ordered pairing function on ordinals to ordinals, and that liberally define sets of ordinals after any formula. Thus it can define relations between ordinals as sets of ordinals that are ordered pairs of ordinals. The ordinals are not limited to the naturals, they are extended by axioms of infinity, size and successor cardinals, so that it results in defining all kinds of ordinals that $\small \sf ZFC$ can define. The idea is that the known set theory $\small \sf ZFC$ can be interpreted in this system via building $\sf L$ inside it. Now I'm wondering about if this extended kind of second order ordinal arithmetic can be extended further to the same length $\small \sf ZFC$ set theory can be extended. Can for example the Wholeness axiom be expressed and have the same strength as it is with extending set theory? Thereby playing the same foundation role extensions of $\small \sf ZFC$ has as regards encoding large cardinal properties.
Wholeness axiom: there is a non-trivial elementary embedding $j$ from the set of all ordinals to itself, with respect to formulas that doesn't mention $j$.
If we assume that $\small \sf 2 oO A$ interprets $\small \sf ZFC$, then would $\small \sf 2oOA$ plus that wholeness axiom be equi-interpretable with $\small \sf ZFC$+ Wholeness axiom?