const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y term inj = \x:set.\y:set.\f:set set.(!z:set.z iIn x -> f z iIn y) & !z:set.z iIn x -> !w:set.w iIn x -> f z = f w -> z = w const setminus : set set set axiom setminusE1: !x:set.!y:set.!z:set.z iIn setminus x y -> z iIn x const inv : set (set set) set set axiom inj_linv: !x:set.!f:set set.(!y:set.y iIn x -> !z:set.z iIn x -> f y = f z -> y = z) -> !y:set.y iIn x -> inv x f (f y) = y const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P var x:set var y:set var f:set set var f2:set set var z:set var w:set var u:set var v:set hyp !x2:set.x2 iIn y -> !y2:set.y2 iIn y -> f2 x2 = f2 y2 -> x2 = y2 hyp (\x2:set.Repl (setminus y (Repl (setminus x x2) \y2:set.f y2)) \y2:set.f2 y2) z = z hyp w = f2 v hyp v iIn y hyp u iIn z claim u iIn Repl (setminus y (Repl (setminus x z) f)) f2 -> inv y f2 w = inv y f2 u -> w = u