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 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 const inv : set (set set) set set lemma !x:set.!y:set.!f:set set.!f2:set set.!z:set.!w:set.!u:set.(!v:set.v iIn y -> !x2:set.x2 iIn y -> f2 v = f2 x2 -> v = x2) -> u iIn setminus y (Repl (setminus x z) f) -> w = f2 u -> u iIn y -> inv y f2 w iIn y var x:set var y:set var f:set set var f2:set set var z:set var w:set hyp !u:set.u iIn y -> !v:set.v iIn y -> f2 u = f2 v -> u = v hyp (\u:set.Repl (setminus y (Repl (setminus x u) \v:set.f v)) \v:set.f2 v) z = z hyp w iIn z claim w iIn Repl (setminus y (Repl (setminus x z) f)) f2 -> inv y f2 w iIn y