const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const proj1 : set set const setsum : set set set const ordsucc : set set const Empty : set axiom proj1E: !x:set.!y:set.y iIn proj1 x -> setsum (ordsucc Empty) y iIn x const ap : set set set axiom apI: !x:set.!y:set.!z:set.setsum y z iIn x -> z iIn ap x y axiom apE: !x:set.!y:set.!z:set.z iIn ap x y -> setsum y z iIn x axiom proj1I: !x:set.!y:set.setsum (ordsucc Empty) y iIn x -> y iIn proj1 x axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y claim !x:set.proj1 x = ap x (ordsucc Empty)