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 term nIn = \x:set.\y:set.~ x iIn y const ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x lemma !x:set.!y:set.Subq x y -> x = ordsucc y -> y iIn x -> y iIn y var x:set var f:set set var y:set var z:set hyp !w:set.w iIn ordsucc (ordsucc x) -> !u:set.u iIn ordsucc (ordsucc x) -> f w = f u -> w = u hyp y iIn ordsucc (ordsucc x) hyp f y = x hyp Subq y z hyp ordsucc z iIn ordsucc (ordsucc x) hyp f (ordsucc z) = x claim y = ordsucc z -> z iIn z