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 const Power : set set axiom setminus_In_Power: !x:set.!y:set.setminus x y iIn Power x const Repl : set (set set) set axiom image_In_Power: !x:set.!y:set.!f:set set.(!z:set.z iIn x -> f z iIn y) -> !z:set.z iIn Power x -> Repl z f iIn Power y lemma !x:set.!y:set.!f:set set.!f2: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) -> (!z:set.z iIn y -> f2 z iIn x) -> (!z:set.z iIn y -> !w:set.w iIn y -> f2 z = f2 w -> z = w) -> (!z:set.z iIn Power x -> Repl (setminus y (Repl (setminus x z) f)) f2 iIn Power x) -> equip x y claim !x:set.!y:set.!f:set set.!f2:set set.inj x y f -> inj y x f2 -> equip x y