While Intuitionistic (or constructive) set Theory IST has received a certain attention from mathematical logicians, so far as I am aware no book providing a systematic introduction to the subject has yet been published.
In particular, proofs of relative consistency with IST for mathematical assertions have been (impl.
This may be the case in part because, as a form of higher-order Intuitionistic logic - the internal logic of a topos - IST has been chiefly developed in a tops-theoretic context.
While Intuitionistic (or constructive) set Theory IST has received a certain attention from mathematical logicians, so far as I am aware no book providing a systematic introduction to the subject has yet been published