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 ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_In_omega: !x:set.x iIn omega -> !y:set.y iIn omega -> x + y iIn omega const SNo : set prop const SNoLev : set set axiom add_SNo_Lev_bd: !x:set.!y:set.SNo x -> SNo y -> Subq (SNoLev (x + y)) (SNoLev x + SNoLev y) axiom In_irref: !x:set.nIn x x axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x var x:set var y:set hyp SNoLev x iIn omega hyp SNo x hyp SNoLev y iIn omega hyp SNo y claim ordinal (SNoLev (x + y)) -> SNoLev (x + y) iIn omega