On a slight weakening of Kripke-Platek Set Theory

The weak set theory $\mathsf{R}\mathsf{e}\mathsf{R}$ is obtained from Kripke-Platek Set Theory ($\mathsf{K}\mathsf{P}$) by replacing the bounded collection scheme with the bounded replacement scheme. We show that $\mathsf{R}\mathsf{e}\mathsf{R}$ proves $\mathsf{T}\mathsf{C}\mathsf{o}$, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to $\mathsf{R}\mathsf{e}\mathsf{R}$ and $\mathsf{K}\mathsf{P}$ have the same consequences. Our proof of $\mathsf{T}\mathsf{C}\mathsf{o}$ relies on the availability of a fragment of class foundation in $\mathsf{R}\mathsf{e}\mathsf{R}$. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of $\mathsf{Z}\mathsf{F}$ that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which $\mathsf{T}\mathsf{C}\mathsf{o}$ fails.