Absolute formula

Question

I'm trying to study the model of Set Theory and I have a problem. If I know that a formula is absolute for a class, could I infer that the formula is true in my class? Namely, I'm trying to prove in $ZF^{-}$ (ZF without the axiom of foundation) that $(V_\omega,\epsilon) \vDash ZF-Inf+ \neg Inf$. I know that union, pairing,.. are $\Delta_0$ formulas that for transitive class are absolute. Can i directly conclude for the previous observation that these axioms are satisfies in $(V_\omega,\epsilon)$? Or I have to prove it in another way?