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 binunion : set set set const Repl : set (set set) set const SetAdjoin : set set set const Sing : set set const ordsucc : set set const Empty : set term SNoElts_ = \x:set.binunion x (Repl x \y:set.SetAdjoin y (Sing (ordsucc Empty))) term SNo_ = \x:set.\y:set.Subq y (SNoElts_ x) & !z:set.z iIn x -> ~(SetAdjoin z (Sing (ordsucc Empty)) iIn y <-> z iIn y) axiom exactly1of2_or: !P:prop.!Q:prop.~(P <-> Q) -> P | Q const ordinal : set prop lemma !x:set.!y:set.!z:set.ordinal y -> Subq x (SNoElts_ y) -> ordinal z -> (\w:set.SetAdjoin w (Sing (ordsucc Empty))) z iIn x -> SetAdjoin z (Sing (ordsucc Empty)) iIn binunion y (Repl y \w:set.SetAdjoin w (Sing (ordsucc Empty))) -> z iIn y lemma !x:set.!y:set.!z:set.Subq x (SNoElts_ y) -> ordinal z -> z iIn x -> z iIn binunion y (Repl y \w:set.SetAdjoin w (Sing (ordsucc Empty))) -> z iIn y var x:set var y:set var z:set var w:set hyp ordinal y hyp ordinal z hyp w iIn y hyp !u:set.u iIn y -> ~(SetAdjoin u (Sing (ordsucc Empty)) iIn x <-> u iIn x) hyp Subq x (SNoElts_ z) claim ordinal w -> w iIn z