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 const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x axiom In_irref: !x:set.nIn x x const ordsucc : set set axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x axiom FalseE: ~ False axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x lemma !x:set.!y:set.!z:set.!w:set.SNo y -> SNo w -> z iIn ordsucc (SNoLev (w + y)) -> ordinal z -> Subq (SNoLev x + SNoLev y) z -> Subq (SNoLev (w + y)) (SNoLev w + SNoLev y) -> SNoLev w + SNoLev y iIn SNoLev x + SNoLev y -> ~ Subq z (SNoLev (w + y)) var x:set var y:set var z:set var w:set hyp SNo y hyp ordinal (SNoLev x + SNoLev 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 (SNoLev w + SNoLev y) (SNoLev x + SNoLev y)