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 TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term nIn = \x:set.\y:set.~ x iIn y axiom In_irref: !x:set.nIn x x const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLev : set set const ordsucc : set set const SNo : set prop var x:set var y:set var z:set var w:set hyp SNo y hyp SNo w hyp z iIn ordsucc (SNoLev (w + y)) hyp ordinal z hyp Subq (SNoLev x + SNoLev y) z hyp Subq (SNoLev (w + y)) (SNoLev w + SNoLev y) hyp SNoLev w + SNoLev y iIn SNoLev x + SNoLev y claim ~ Subq z (SNoLev (w + y))