ITP001 Axioms: ITP027^7.ax
%------------------------------------------------------------------------------
% File : ITP027^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : pred_set.ax [Gau19]
% : HL4027^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 706 ( 175 unt; 114 typ; 0 def)
% Number of atoms : 2599 ( 463 equ; 107 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 8049 ( 107 ~; 29 |; 268 &;7130 @)
% ( 173 <=>; 342 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg;7130 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 1587 (1587 >; 0 *; 0 +; 0 <<)
% Number of symbols : 114 ( 112 usr; 5 con; 0-6 aty)
% Number of variables : 2336 ( 68 ^2087 !; 80 ?;2336 :)
% ( 101 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Earithmetic_2EABS__DIFF,type,
c_2Earithmetic_2EABS__DIFF: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Epred__set_2EBIGINTER,type,
c_2Epred__set_2EBIGINTER:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIJ,type,
c_2Epred__set_2EBIJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2ECOMPL,type,
c_2Epred__set_2ECOMPL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epred__set_2ECROSS,type,
c_2Epred__set_2ECROSS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDFUNSET,type,
c_2Epred__set_2EDFUNSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27b > $o ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EDIV,type,
c_2Earithmetic_2EDIV: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Erelation_2EEMPTY__REL,type,
c_2Erelation_2EEMPTY__REL:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Earithmetic_2EEXP,type,
c_2Earithmetic_2EEXP: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Earithmetic_2EFUNPOW,type,
c_2Earithmetic_2EFUNPOW:
!>[A_27a: $tType] : ( ( A_27a > A_27a ) > tyop_2Enum_2Enum > A_27a > A_27a ) ).
thf(c_2Epred__set_2EFUNSET,type,
c_2Epred__set_2EFUNSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINJ,type,
c_2Epred__set_2EINJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Esum_2EINL,type,
c_2Esum_2EINL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EINR,type,
c_2Esum_2EINR:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EITSET,type,
c_2Epred__set_2EITSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > ( A_27a > $o ) > A_27b > A_27b ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Epred__set_2ELINV,type,
c_2Epred__set_2ELINV:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > A_27a ) ).
thf(c_2Epred__set_2ELINV__OPT,type,
c_2Epred__set_2ELINV__OPT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Earithmetic_2EMAX,type,
c_2Earithmetic_2EMAX: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EMAX__SET,type,
c_2Epred__set_2EMAX__SET: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EMIN,type,
c_2Earithmetic_2EMIN: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EMIN__SET,type,
c_2Epred__set_2EMIN__SET: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EPOW,type,
c_2Epred__set_2EPOW:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EPREIMAGE,type,
c_2Epred__set_2EPREIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27b > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EPROD__IMAGE,type,
c_2Epred__set_2EPROD__IMAGE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Epred__set_2EPROD__SET,type,
c_2Epred__set_2EPROD__SET: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2EPSUBSET,type,
c_2Epred__set_2EPSUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EREL__RESTRICT,type,
c_2Epred__set_2EREL__RESTRICT:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EREST,type,
c_2Epred__set_2EREST:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2ERINV,type,
c_2Epred__set_2ERINV:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > A_27a ) ).
thf(c_2Erelation_2ERSUBSET,type,
c_2Erelation_2ERSUBSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2ESING,type,
c_2Epred__set_2ESING:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ESUM__IMAGE,type,
c_2Epred__set_2ESUM__IMAGE:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Epred__set_2ESUM__SET,type,
c_2Epred__set_2ESUM__SET: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ESURJ,type,
c_2Epred__set_2ESURJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Erelation_2EStrongOrder,type,
c_2Erelation_2EStrongOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Erelation_2ETC,type,
c_2Erelation_2ETC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Eoption_2ETHE,type,
c_2Eoption_2ETHE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27a ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ecombin_2EUPDATE,type,
c_2Ecombin_2EUPDATE:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Epred__set_2Echooser,type,
c_2Epred__set_2Echooser:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Epred__set_2Ecount,type,
c_2Epred__set_2Ecount: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Epred__set_2Ecountable,type,
c_2Epred__set_2Ecountable:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2Eenumerate,type,
c_2Epred__set_2Eenumerate:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Epred__set_2Eequiv__on,type,
c_2Epred__set_2Eequiv__on:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Einv,type,
c_2Erelation_2Einv:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27b > A_27a > $o ) ).
thf(c_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2Eis__measure__maximal,type,
c_2Epred__set_2Eis__measure__maximal:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Enumpair_2Enfst,type,
c_2Enumpair_2Enfst: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enumpair_2Enpair,type,
c_2Enumpair_2Enpair: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enumpair_2Ensnd,type,
c_2Enumpair_2Ensnd: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2Enum__to__pair,type,
c_2Epred__set_2Enum__to__pair: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Epred__set_2Epair__to__num,type,
c_2Epred__set_2Epair__to__num: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2Epairwise,type,
c_2Epred__set_2Epairwise:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2Epartition,type,
c_2Epred__set_2Epartition:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2Eschroeder__close,type,
c_2Epred__set_2Eschroeder__close:
!>[A_27a: $tType] : ( ( A_27a > A_27a ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ),V1v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1v @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b @ V0f ) )
<=> ? [V2x: A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ $o @ V1v @ c_2Ebool_2ET )
= ( V0f @ V2x ) ) ) ).
thf(thm_2Epred__set_2EEMPTY__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( ^ [V0x: A_27a] : c_2Ebool_2EF ) ) ).
thf(thm_2Epred__set_2EUNIV__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EUNIV @ A_27a )
= ( ^ [V0x: A_27a] : c_2Ebool_2ET ) ) ).
thf(thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
=> ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EPSUBSET__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V1t )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( (~) @ ( V0s = V1t ) ) ) ) ).
thf(thm_2Epred__set_2EUNION__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_5C_2F @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ) ).
thf(thm_2Epred__set_2EINTER__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V1t )
<=> ( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EDIFF__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EINSERT__DEF,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2y @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ A_27a @ V2y @ V0x ) @ ( c_2Ebool_2EIN @ A_27a @ V2y @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EDELETE__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V1x )
= ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2ECHOICE__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( c_2Ebool_2EIN @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) @ V0s ) ) ).
thf(thm_2Epred__set_2EREST__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EREST @ A_27a @ V0s )
= ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2ESING__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ESING @ A_27a @ V0s )
<=> ? [V1x: A_27a] :
( V0s
= ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V0f @ V2x ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
<=> ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2t ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V1s ) )
=> ( ( ( V0f @ V4x )
= ( V0f @ V5y ) )
=> ( V4x = V5y ) ) ) ) ) ).
thf(thm_2Epred__set_2ESURJ__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
<=> ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2t ) )
& ! [V4x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4x @ V2t )
=> ? [V5y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5y @ V1s )
& ( ( V0f @ V5y )
= V4x ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIJ__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
<=> ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
& ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2EFUNSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o] :
( ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0P @ V1Q )
= ( ^ [V2f: A_27a > A_27b] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V0P ) @ ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ V1Q ) ) ) ) ) ).
thf(thm_2Epred__set_2EDFUNSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27a > A_27b > $o] :
( ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b @ V0P @ V1Q )
= ( ^ [V2f: A_27a > A_27b] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V0P ) @ ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ ( V1Q @ V3x ) ) ) ) ) ) ).
thf(thm_2Epred__set_2Eschroeder__close__def,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1s: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2Eschroeder__close @ A_27a @ V0f @ V1s @ V2x )
<=> ? [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Earithmetic_2EFUNPOW @ ( A_27a > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V0f ) @ V3n @ V1s ) ) ) ).
thf(thm_2Epred__set_2ELINV__OPT__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2y: A_27b] :
( ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b @ V0f @ V1s @ V2y )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ebool_2EIN @ A_27b @ V2y @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) )
@ ( c_2Eoption_2ESOME @ A_27a
@ ( c_2Emin_2E_40 @ A_27a
@ ^ [V3x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V3x ) @ V2y ) ) ) )
@ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ELINV__LO,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2y: A_27b] :
( ( c_2Epred__set_2ELINV @ A_27a @ A_27b @ V0f @ V1s @ V2y )
= ( c_2Eoption_2ETHE @ A_27a @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b @ V0f @ V1s @ V2y ) ) ) ).
thf(thm_2Epred__set_2ERINV__LO,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2y: A_27b] :
( ( c_2Epred__set_2ERINV @ A_27a @ A_27b @ V0f @ V1s @ V2y )
= ( c_2Eoption_2ETHE @ A_27a @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b @ V0f @ V1s @ V2y ) ) ) ).
thf(thm_2Epred__set_2EFINITE__DEF,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
<=> ! [V1P: ( A_27a > $o ) > $o] :
( ( ( V1P @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2s: A_27a > $o] :
( ( V1P @ V2s )
=> ! [V3e: A_27a] : ( V1P @ ( c_2Epred__set_2EINSERT @ A_27a @ V3e @ V2s ) ) ) )
=> ( V1P @ V0s ) ) ) ).
thf(thm_2Epred__set_2EREL__RESTRICT__DEF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ V1s @ V2x @ V3y )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( V0R @ V2x @ V3y ) ) ) ).
thf(thm_2Epred__set_2ECARD__DEF,axiom,
! [A_27a: $tType] :
( ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= c_2Enum_2E0 )
& ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1x: A_27a] :
( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V0s ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Enum_2ESUC @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ) ) ).
thf(thm_2Epred__set_2Ecount__def,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Ecount @ V0n )
= ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1m: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1m @ ( c_2Eprim__rec_2E_3C @ V1m @ V0n ) ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ V0P )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V1x
@ ( c_2Ebool_2E_3F @ ( A_27a > $o )
@ ^ [V2s: A_27a > $o] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V0P ) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V2s ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIGINTER,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ V0P )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V1x
@ ( c_2Ebool_2E_21 @ ( A_27a > $o )
@ ^ [V2s: A_27a > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V0P ) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V2s ) ) ) ) ) ) ).
thf(thm_2Epred__set_2ECROSS__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V2p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ V2p @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2p ) @ V0P ) @ ( c_2Ebool_2EIN @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2p ) @ V1Q ) ) ) ) ) ).
thf(thm_2Epred__set_2ECOMPL__DEF,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2ECOMPL @ A_27a @ V0P )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0P ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__DEF,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s )
= ( c_2Epred__set_2EITSET @ A_27a @ tyop_2Enum_2Enum
@ ^ [V2e: A_27a,V3acc: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( V0f @ V2e ) @ V3acc )
@ V1s
@ c_2Enum_2E0 ) ) ).
thf(thm_2Epred__set_2ESUM__SET__DEF,axiom,
( c_2Epred__set_2ESUM__SET
= ( c_2Epred__set_2ESUM__IMAGE @ tyop_2Enum_2Enum @ ( c_2Ecombin_2EI @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Epred__set_2EPROD__IMAGE__DEF,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EPROD__IMAGE @ A_27a @ V0f @ V1s )
= ( c_2Epred__set_2EITSET @ A_27a @ tyop_2Enum_2Enum
@ ^ [V2e: A_27a,V3acc: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A @ ( V0f @ V2e ) @ V3acc )
@ V1s
@ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Epred__set_2EPROD__SET__DEF,axiom,
( c_2Epred__set_2EPROD__SET
= ( c_2Epred__set_2EPROD__IMAGE @ tyop_2Enum_2Enum @ ( c_2Ecombin_2EI @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Epred__set_2EMAX__SET__DEF,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EMAX__SET @ V0s ) @ V0s )
& ! [V1y: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1y @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ V1y @ ( c_2Epred__set_2EMAX__SET @ V0s ) ) ) ) )
& ( ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
=> ( ( c_2Epred__set_2EMAX__SET @ V0s )
= c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Epred__set_2EMIN__SET__DEF,axiom,
c_2Epred__set_2EMIN__SET = c_2Ewhile_2ELEAST ).
thf(thm_2Epred__set_2EPOW__DEF,axiom,
! [A_27a: $tType,V0set: A_27a > $o] :
( ( c_2Epred__set_2EPOW @ A_27a @ V0set )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1s @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V0set ) ) ) ) ).
thf(thm_2Epred__set_2Eequiv__on__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2Eequiv__on @ A_27a @ V0R @ V1s )
<=> ( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( V0R @ V2x @ V2x ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V1s ) )
=> ( ( V0R @ V3x @ V4y )
= ( V0R @ V4y @ V3x ) ) )
& ! [V5x: A_27a,V6y: A_27a,V7z: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V6y @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V7z @ V1s )
& ( V0R @ V5x @ V6y )
& ( V0R @ V6y @ V7z ) )
=> ( V0R @ V5x @ V7z ) ) ) ) ).
thf(thm_2Epred__set_2Epartition__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2Epartition @ A_27a @ V0R @ V1s )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2t: A_27a > $o] :
( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V2t
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V3x: A_27a] :
( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
@ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V2t
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4y @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4y @ V1s ) @ ( V0R @ V3x @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2Epred__set_2Epairwise__def,axiom,
! [A_27a: $tType,V0P: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2Epairwise @ A_27a @ V0P @ V1s )
<=> ! [V2e1: A_27a,V3e2: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2e1 @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3e2 @ V1s ) )
=> ( V0P @ V2e1 @ V3e2 ) ) ) ).
thf(thm_2Epred__set_2Echooser__def,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2Echooser @ A_27a @ V0s @ c_2Enum_2E0 )
= ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) )
& ! [V1s: A_27a > $o,V2n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Echooser @ A_27a @ V1s @ ( c_2Enum_2ESUC @ V2n ) )
= ( c_2Epred__set_2Echooser @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V1s ) @ V2n ) ) ) ).
thf(thm_2Epred__set_2Ecountable__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
<=> ? [V1f: A_27a > tyop_2Enum_2Enum] : ( c_2Epred__set_2EINJ @ A_27a @ tyop_2Enum_2Enum @ V1f @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Epred__set_2Enum__to__pair__def,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Enum__to__pair @ V0n )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Enumpair_2Enfst @ V0n ) @ ( c_2Enumpair_2Ensnd @ V0n ) ) ) ).
thf(thm_2Epred__set_2Epair__to__num__def,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Epair__to__num @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0m @ V1n ) )
= ( c_2Enumpair_2Enpair @ V0m @ V1n ) ) ).
thf(thm_2Epred__set_2Eenumerate__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Eenumerate @ A_27a @ V0s )
= ( c_2Emin_2E_40 @ ( tyop_2Enum_2Enum > A_27a )
@ ^ [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V2x ) @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2Eis__measure__maximal__def,axiom,
! [A_27a: $tType,V0m: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2Eis__measure__maximal @ A_27a @ V0m @ V1s @ V2x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ! [V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0m @ V3y ) @ ( V0m @ V2x ) ) ) ) ) ).
thf(thm_2Epred__set_2ESPECIFICATION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0P )
= ( V0P @ V1x ) ) ).
thf(thm_2Epred__set_2EIN__APP,axiom,
! [A_27a: $tType,V0x: A_27a,V1P: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1P )
= ( V1P @ V0x ) ) ).
thf(thm_2Epred__set_2EIN__ABS,axiom,
! [A_27a: $tType,V0x: A_27a,V1P: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x
@ ^ [V2x: A_27a] : ( V1P @ V2x ) )
= ( V1P @ V0x ) ) ).
thf(thm_2Epred__set_2EEXTENSION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( V0s = V1t )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
= ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2ENOT__EQUAL__SETS,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( (~) @ ( V0s = V1t ) )
<=> ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t )
<=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ENUM__SET__WOP,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1n @ V0s )
<=> ? [V2n: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2n @ V0s )
& ! [V3m: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3m @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ V2n @ V3m ) ) ) ) ).
thf(thm_2Epred__set_2EGSPECIFICATION__applied,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ),V1v: A_27a] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b @ V0f @ V1v )
<=> ? [V2x: A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ $o @ V1v @ c_2Ebool_2ET )
= ( V0f @ V2x ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__ETA,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V1x @ ( V0P @ V1x ) ) )
= V0P ) ).
thf(thm_2Epred__set_2EGSPEC__PAIR__ETA,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V1x: A_27a,V2y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) @ ( V0P @ V1x @ V2y ) ) ) )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o @ V0P ) ) ).
thf(thm_2Epred__set_2EIN__GSPEC__IFF,axiom,
! [A_27a: $tType,V0y: A_27a,V1P: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0y
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( V1P @ V2x ) ) ) )
= ( V1P @ V0y ) ) ).
thf(thm_2Epred__set_2EPAIR__IN__GSPEC__IFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2P: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V3x: A_27a,V4y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3x @ V4y ) @ ( V2P @ V3x @ V4y ) ) ) ) )
= ( V2P @ V1x @ V0y ) ) ).
thf(thm_2Epred__set_2EIN__GSPEC,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1y: A_27a,V2x: A_27b,V3P: A_27a > $o] :
( ( ( V3P @ V1y )
& ( V2x
= ( V0f @ V1y ) ) )
=> ( c_2Ebool_2EIN @ A_27b @ V2x
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V0f @ V4x ) @ ( V3P @ V4x ) ) ) ) ) ).
thf(thm_2Epred__set_2EPAIR__IN__GSPEC__1,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1b: A_27b,V2a: A_27a,V3P: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2a @ V1b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a
@ ^ [V4y: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V4y @ V0x ) @ ( V3P @ V4y ) ) ) )
<=> ( ( V3P @ V2a )
& ( V1b = V0x ) ) ) ).
thf(thm_2Epred__set_2EPAIR__IN__GSPEC__2,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1b: A_27b,V2a: A_27a,V3P: A_27b > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2a @ V1b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b
@ ^ [V4y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V4y ) @ ( V3P @ V4y ) ) ) )
<=> ( ( V3P @ V1b )
& ( V2a = V0x ) ) ) ).
thf(thm_2Epred__set_2EPAIR__IN__GSPEC__same,axiom,
! [A_27a: $tType,V0b: A_27a,V1a: A_27a,V2P: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1a @ V0b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x @ V3x ) @ ( V2P @ V3x ) ) ) )
<=> ( ( V2P @ V1a )
& ( V1a = V0b ) ) ) ).
thf(thm_2Epred__set_2ESET__MINIMUM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1M: A_27a > tyop_2Enum_2Enum] :
( ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
<=> ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
& ! [V4y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4y @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V1M @ V3x ) @ ( V1M @ V4y ) ) ) ) ) ).
thf(thm_2Epred__set_2ENOT__IN__EMPTY,axiom,
! [A_27a: $tType,V0x: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EMEMBER__NOT__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ? [V1x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
<=> ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EEMPTY__applied,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2EEMPTY @ A_27a @ V0x )
= c_2Ebool_2EF ) ).
thf(thm_2Epred__set_2EIN__UNIV,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2EUNIV__applied,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Epred__set_2EUNIV @ A_27a @ V0x ) ).
thf(thm_2Epred__set_2EUNIV__NOT__EMPTY,axiom,
! [A_27a: $tType] :
( (~)
@ ( ( c_2Epred__set_2EUNIV @ A_27a )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EEMPTY__NOT__UNIV,axiom,
! [A_27a: $tType] :
( (~)
@ ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EEQ__UNIV,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ! [V1x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
<=> ( V0s
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EIN__EQ__UNIV__IMP,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( V0s
= ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ! [V1v: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V1v @ V0s ) ) ).
thf(thm_2Epred__set_2ESUBSET__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0P @ V1Q )
=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0P )
=> ( c_2Ebool_2EIN @ A_27a @ V2x @ V1Q ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__applied,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
<=> ! [V2x: A_27a] :
( ( V0s @ V2x )
=> ( V1t @ V2x ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__TRANS,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V2u ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2ESUBSET__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V0s ) ).
thf(thm_2Epred__set_2ESUBSET__ANTISYM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) )
=> ( V0s = V1t ) ) ).
thf(thm_2Epred__set_2EEMPTY__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0s ) ).
thf(thm_2Epred__set_2ESUBSET__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__UNIV,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2EUNIV__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0s )
<=> ( V0s
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EEQ__SUBSET__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( V0s = V1t )
=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__SUBSET__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) )
<=> ( V0s = V1t ) ) ).
thf(thm_2Epred__set_2ESUBSET__ADD,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a > $o,V1n: tyop_2Enum_2Enum,V2d: tyop_2Enum_2Enum] :
( ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V0f @ V3n ) @ ( V0f @ ( c_2Enum_2ESUC @ V3n ) ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( V0f @ V1n ) @ ( V0f @ ( c_2Earithmetic_2E_2B @ V1n @ V2d ) ) ) ) ).
thf(thm_2Epred__set_2EK__SUBSET,axiom,
! [A_27a: $tType,V0x: $o,V1y: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Ecombin_2EK @ $o @ A_27a @ V0x ) @ V1y )
<=> ( ( (~) @ V0x )
| ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V1y ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__K,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1y: $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0x @ ( c_2Ecombin_2EK @ $o @ A_27a @ V1y ) )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
| V1y ) ) ).
thf(thm_2Epred__set_2EPSUBSET__TRANS,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2EPSUBSET @ A_27a @ V1t @ V2u ) )
=> ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2EPSUBSET__IRREFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( (~) @ ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V0s ) ) ).
thf(thm_2Epred__set_2ENOT__PSUBSET__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( (~) @ ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ENOT__UNIV__PSUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( (~) @ ( c_2Epred__set_2EPSUBSET @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0s ) ) ).
thf(thm_2Epred__set_2EPSUBSET__UNIV,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) )
<=> ? [V1x: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) ) ) ).
thf(thm_2Epred__set_2EIN__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
| ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EUNION__applied,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t @ V2x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
| ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EUNION__ASSOC,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ ( c_2Epred__set_2EUNION @ A_27a @ V1t @ V2u ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ V2u ) ) ).
thf(thm_2Epred__set_2EUNION__IDEMPOT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V0s )
= V0s ) ).
thf(thm_2Epred__set_2EUNION__COMM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EUNION @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Epred__set_2ESUBSET__UNION,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o,V1t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) )
& ! [V2s: A_27a > $o,V3t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V2s @ ( c_2Epred__set_2EUNION @ A_27a @ V3t @ V2s ) ) ) ).
thf(thm_2Epred__set_2EUNION__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ V2u )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V2u )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V2u ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__UNION__ABSORPTION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
<=> ( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t )
= V1t ) ) ).
thf(thm_2Epred__set_2EUNION__EMPTY,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0s )
= V0s )
& ! [V1s: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V1s @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V1s ) ) ).
thf(thm_2Epred__set_2EUNION__UNIV,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0s )
= ( c_2Epred__set_2EUNIV @ A_27a ) )
& ! [V1s: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V1s @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EEMPTY__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EFORALL__IN__UNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
=> ( V0P @ V3x ) )
<=> ( ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s )
=> ( V0P @ V4x ) )
& ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ V2t )
=> ( V0P @ V5x ) ) ) ) ).
thf(thm_2Epred__set_2EIN__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EINTER__applied,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t @ V2x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ).
thf(thm_2Epred__set_2EINTER__ASSOC,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V2u ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ V2u ) ) ).
thf(thm_2Epred__set_2EINTER__IDEMPOT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V0s )
= V0s ) ).
thf(thm_2Epred__set_2EINTER__COMM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Epred__set_2EINTER__SUBSET,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o,V1t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ V0s )
& ! [V2s: A_27a > $o,V3t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V3t @ V2s ) @ V2s ) ) ).
thf(thm_2Epred__set_2ESUBSET__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V2u ) )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V2u ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__INTER__ABSORPTION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
<=> ( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= V0s ) ) ).
thf(thm_2Epred__set_2ESUBSET__INTER1,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
=> ( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= V0s ) ) ).
thf(thm_2Epred__set_2ESUBSET__INTER2,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
=> ( ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V0s )
= V0s ) ) ).
thf(thm_2Epred__set_2EINTER__EMPTY,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V1s: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V1s @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EINTER__UNIV,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0s )
= V0s )
& ! [V1s: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V1s @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V1s ) ) ).
thf(thm_2Epred__set_2EUNION__OVER__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ ( c_2Epred__set_2EUNION @ A_27a @ V1t @ V2u ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V2u ) ) ) ).
thf(thm_2Epred__set_2EINTER__OVER__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V2u ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V2u ) ) ) ).
thf(thm_2Epred__set_2EIN__DISJOINT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V1t )
<=> ( (~)
@ ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__SYM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EDISJOINT @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Epred__set_2EDISJOINT__ALT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V1t )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0s )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__EMPTY__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V0s ) ) ).
thf(thm_2Epred__set_2EDISJOINT__EMPTY__REFL__RWT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V0s )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ V2u )
<=> ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V2u )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1t @ V2u ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__UNION__BOTH,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ V2u )
<=> ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V2u )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1t @ V2u ) ) )
& ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V2u @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V2u )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1t @ V2u ) ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2u @ V1t ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2EIN__DIFF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EDIFF__applied,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t @ V2x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EDIFF__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V0s ) ).
thf(thm_2Epred__set_2EEMPTY__DIFF,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EDIFF__UNIV,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EDIFF__DIFF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) @ V1t )
= ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2EDIFF__EQ__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EDIFF__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) @ V0s ) ).
thf(thm_2Epred__set_2EUNION__DIFF,axiom,
! [A_27a: $tType,V0t: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V0t )
=> ( ( ( c_2Epred__set_2EUNION @ A_27a @ V1s @ ( c_2Epred__set_2EDIFF @ A_27a @ V0t @ V1s ) )
= V0t )
& ( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0t @ V1s ) @ V1s )
= V0t ) ) ) ).
thf(thm_2Epred__set_2EDIFF__DIFF__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s )
=> ( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) )
= V1t ) ) ).
thf(thm_2Epred__set_2EDIFF__UNION,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1y: A_27a > $o,V2z: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0x @ ( c_2Epred__set_2EUNION @ A_27a @ V1y @ V2z ) )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0x @ V1y ) @ V2z ) ) ).
thf(thm_2Epred__set_2EDIFF__COMM,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1y: A_27a > $o,V2z: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0x @ V1y ) @ V2z )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0x @ V2z ) @ V1y ) ) ).
thf(thm_2Epred__set_2EDIFF__SAME__UNION,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1y: A_27a > $o] :
( ( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0x @ V1y ) @ V0x )
= ( c_2Epred__set_2EDIFF @ A_27a @ V1y @ V0x ) )
& ( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0x @ V1y ) @ V1y )
= ( c_2Epred__set_2EDIFF @ A_27a @ V0x @ V1y ) ) ) ).
thf(thm_2Epred__set_2EDIFF__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2g: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) @ V2g )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V2g ) @ V1t ) ) ).
thf(thm_2Epred__set_2EDIFF__INTER2,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ ( c_2Epred__set_2EINTER @ A_27a @ V1t @ V0s ) )
= ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2EDISJOINT__DIFF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V1t @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) @ V1t ) ) ).
thf(thm_2Epred__set_2EDISJOINT__DIFFS,axiom,
! [A_27a: $tType,V0g: tyop_2Enum_2Enum > A_27a > $o,V1f: tyop_2Enum_2Enum > A_27a > $o,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ! [V4n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( V1f @ V4n ) @ ( V1f @ ( c_2Enum_2ESUC @ V4n ) ) )
& ! [V5n: tyop_2Enum_2Enum] :
( ( V0g @ V5n )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( V1f @ ( c_2Enum_2ESUC @ V5n ) ) @ ( V1f @ V5n ) ) )
& ( (~) @ ( V2m = V3n ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V0g @ V2m ) @ ( V0g @ V3n ) ) ) ).
thf(thm_2Epred__set_2EIN__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ V2s ) )
<=> ( ( V0x = V1y )
| ( c_2Ebool_2EIN @ A_27a @ V0x @ V2s ) ) ) ).
thf(thm_2Epred__set_2EINSERT__applied,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ V2s @ V0x )
<=> ( ( V0x = V1y )
| ( c_2Ebool_2EIN @ A_27a @ V0x @ V2s ) ) ) ).
thf(thm_2Epred__set_2ECOMPONENT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] : ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) ) ).
thf(thm_2Epred__set_2ESET__CASES,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V1x: A_27a,V2t: A_27a > $o] :
( ( V0s
= ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V2t ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EDECOMPOSITION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
<=> ? [V2t: A_27a > $o] :
( ( V0s
= ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V2t ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EABSORPTION,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
<=> ( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s )
= V1s ) ) ).
thf(thm_2Epred__set_2EABSORPTION__RWT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
=> ( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s )
= V1s ) ) ).
thf(thm_2Epred__set_2EINSERT__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) ) ).
thf(thm_2Epred__set_2EINSERT__COMM,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ V2s ) )
= ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V2s ) ) ) ).
thf(thm_2Epred__set_2EINSERT__UNIV,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2ENOT__INSERT__EMPTY,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( (~)
@ ( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ENOT__EMPTY__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( (~)
@ ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) ) ) ).
thf(thm_2Epred__set_2EINSERT__UNION,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V2t )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V2t ) @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EINSERT__UNION__EQ,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V2t )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2EINSERT__INTER,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V2t )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V2t ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V2t )
<=> ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s @ V2t )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__INSERT_27,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V2t @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) )
<=> ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V2t @ V1s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EINSERT__SUBSET,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V2t )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V2t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) )
=> ! [V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V2t ) )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2EINSERT__DIFF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V0s ) @ V1t )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EUNIV__BOOL,axiom,
( ( c_2Epred__set_2EUNIV @ $o )
= ( c_2Epred__set_2EINSERT @ $o @ c_2Ebool_2ET @ ( c_2Epred__set_2EINSERT @ $o @ c_2Ebool_2EF @ ( c_2Epred__set_2EEMPTY @ $o ) ) ) ) ).
thf(thm_2Epred__set_2EFORALL__IN__INSERT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a,V2s: A_27a > $o] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2s ) )
=> ( V0P @ V3x ) )
<=> ( ( V0P @ V1a )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
=> ( V0P @ V4x ) ) ) ) ).
thf(thm_2Epred__set_2EEXISTS__IN__INSERT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1a: A_27a,V2s: A_27a > $o] :
( ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2s ) )
& ( V0P @ V3x ) )
<=> ( ( V0P @ V1a )
| ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
& ( V0P @ V4x ) ) ) ) ).
thf(thm_2Epred__set_2EIN__DELETE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2y ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
& ( (~) @ ( V1x = V2y ) ) ) ) ).
thf(thm_2Epred__set_2EDELETE__applied,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2y @ V1x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
& ( (~) @ ( V1x = V2y ) ) ) ) ).
thf(thm_2Epred__set_2EDELETE__NON__ELEMENT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) )
<=> ( ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x )
= V1s ) ) ).
thf(thm_2Epred__set_2EDELETE__NON__ELEMENT__RWT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) )
=> ( ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V1x )
= V0s ) ) ).
thf(thm_2Epred__set_2EIN__DELETE__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a,V2x_27: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
= ( c_2Ebool_2EIN @ A_27a @ V2x_27 @ V0s ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2x_27 ) )
= ( c_2Ebool_2EIN @ A_27a @ V2x_27 @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V1x ) ) ) ) ).
thf(thm_2Epred__set_2EEMPTY__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EELT__IN__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] : ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) ) ) ).
thf(thm_2Epred__set_2EDELETE__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) @ V0x )
= ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) ) ).
thf(thm_2Epred__set_2EDELETE__COMM,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V0x ) @ V1y )
= ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1y ) @ V0x ) ) ).
thf(thm_2Epred__set_2EDELETE__SUBSET,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) @ V1s ) ).
thf(thm_2Epred__set_2ESUBSET__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EDELETE @ A_27a @ V2t @ V0x ) )
<=> ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__INSERT__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V2t ) )
= ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) @ V2t ) ) ).
thf(thm_2Epred__set_2ESUBSET__OF__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) ) ).
thf(thm_2Epred__set_2EDIFF__INSERT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1t ) )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2x ) @ V1t ) ) ).
thf(thm_2Epred__set_2EPSUBSET__INSERT__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V1t )
<=> ? [V2x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V0s ) @ V1t ) ) ) ).
thf(thm_2Epred__set_2EPSUBSET__MEMBER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V1t )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ? [V2y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2y @ V1t )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2y @ V0s ) ) ) ) ) ).
thf(thm_2Epred__set_2EDELETE__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V2s ) @ V1y )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Emin_2E_3D @ A_27a @ V0x @ V1y ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1y ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1y ) ) ) ) ).
thf(thm_2Epred__set_2EINSERT__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
=> ( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) )
= V1s ) ) ).
thf(thm_2Epred__set_2EDELETE__INTER,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2x ) @ V1t )
= ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) @ V2x ) ) ).
thf(thm_2Epred__set_2EDISJOINT__DELETE__SYM,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2x ) @ V1t )
= ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1t @ V2x ) @ V0s ) ) ).
thf(thm_2Epred__set_2EIN__REST,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EREST @ A_27a @ V1s ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
& ( (~)
@ ( V0x
= ( c_2Epred__set_2ECHOICE @ A_27a @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EREST__applied,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EREST @ A_27a @ V1s @ V0x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
& ( (~)
@ ( V0x
= ( c_2Epred__set_2ECHOICE @ A_27a @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2ECHOICE__NOT__IN__REST,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( (~) @ ( c_2Ebool_2EIN @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2ECHOICE__INSERT__REST,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( ( c_2Epred__set_2EINSERT @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) )
= V0s ) ) ).
thf(thm_2Epred__set_2EREST__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) @ V0s ) ).
thf(thm_2Epred__set_2EREST__PSUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( c_2Epred__set_2EPSUBSET @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) @ V0s ) ) ).
thf(thm_2Epred__set_2ESING,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Epred__set_2ESING @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ESING__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2ESING @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= c_2Ebool_2EF ) ).
thf(thm_2Epred__set_2ESING__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2ESING @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) )
<=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V1s
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(thm_2Epred__set_2ESING__UNION,axiom,
! [A_27a: $tType,V0t: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2ESING @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V0t ) )
<=> ( ( ( c_2Epred__set_2ESING @ A_27a @ V1s )
& ( V0t
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
| ( ( c_2Epred__set_2ESING @ A_27a @ V0t )
& ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
| ( ( c_2Epred__set_2ESING @ A_27a @ V1s )
& ( c_2Epred__set_2ESING @ A_27a @ V0t )
& ( V1s = V0t ) ) ) ) ).
thf(thm_2Epred__set_2EIN__SING,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Epred__set_2ESING__applied,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0x )
<=> ( V0x = V1y ) ) ).
thf(thm_2Epred__set_2ENOT__SING__EMPTY,axiom,
! [A_27a: $tType,V0x: A_27a] :
( (~)
@ ( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ENOT__EMPTY__SING,axiom,
! [A_27a: $tType,V0x: A_27a] :
( (~)
@ ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EEQUAL__SING,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Epred__set_2EDISJOINT__SING__EMPTY,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EINSERT__SING__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V0s )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s ) ) ).
thf(thm_2Epred__set_2ESING__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EDELETE__EQ__SING,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s )
=> ( ( ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V1x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0s
= ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(thm_2Epred__set_2ECHOICE__SING,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2ECHOICE @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= V0x ) ).
thf(thm_2Epred__set_2EREST__SING,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2EREST @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2ESING__IFF__EMPTY__REST,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ESING @ A_27a @ V0s )
<=> ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( ( c_2Epred__set_2EREST @ A_27a @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EIN__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V0y @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s ) )
<=> ? [V3x: A_27a] :
( ( V0y
= ( V2f @ V3x ) )
& ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__applied,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s @ V0y )
<=> ? [V3x: A_27a] :
( ( V0y
= ( V2f @ V3x ) )
& ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__IN,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
=> ! [V2f: A_27a > A_27b] : ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V0x ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ).
thf(thm_2Epred__set_2EIMAGE__ID,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a
@ ^ [V1x: A_27a] : V1x
@ V0s )
= V0s ) ).
thf(thm_2Epred__set_2EIMAGE__I,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ ( c_2Ecombin_2EI @ A_27a ) @ V0s )
= V0s ) ).
thf(thm_2Epred__set_2EIMAGE__II,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( A_27a > $o ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__COMPOSE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c,V1g: A_27a > A_27b,V2s: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V0f @ V1g ) @ V2s )
= ( c_2Epred__set_2EIMAGE @ A_27b @ A_27c @ V0f @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1g @ V2s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__INSERT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V2s ) )
= ( c_2Epred__set_2EINSERT @ A_27b @ ( V0f @ V1x ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__EQ__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b] :
( ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__DELETE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a,V2s: A_27a > $o] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V2s ) )
=> ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1x ) )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__UNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Epred__set_2EUNION @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
=> ! [V2f: A_27a > A_27b] : ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V0s ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1t ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__INTER,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) @ ( c_2Epred__set_2EINTER @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2f: A_27a > A_27b] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V2f @ V3x )
= ( V2f @ V4y ) )
<=> ( V3x = V4y ) )
=> ( ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s1 )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V0s2 ) )
<=> ( V1s1 = V0s2 ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2f: A_27a > A_27b] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V2f @ V3x )
= ( V2f @ V4y ) )
<=> ( V3x = V4y ) )
=> ( ( c_2Epred__set_2EDISJOINT @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s1 ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V0s2 ) )
= ( c_2Epred__set_2EDISJOINT @ A_27a @ V1s1 @ V0s2 ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2f_27: A_27a > A_27b,V3s_27: A_27a > $o] :
( ( ( V1s = V3s_27 )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V3s_27 )
=> ( ( V0f @ V4x )
= ( V2f_27 @ V4x ) ) ) )
=> ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f_27 @ V3s_27 ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o )] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b @ V0f )
= ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ ( c_2Ecombin_2Eo @ A_27b @ A_27a @ ( tyop_2Epair_2Eprod @ A_27a @ $o ) @ ( c_2Epair_2EFST @ A_27a @ $o ) @ V0f ) @ ( c_2Ecombin_2Eo @ A_27b @ $o @ ( tyop_2Epair_2Eprod @ A_27a @ $o ) @ ( c_2Epair_2ESND @ A_27a @ $o ) @ V0f ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27a,V2s: A_27c > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EIMAGE @ A_27c @ A_27a @ V1g @ V2s ) )
= ( c_2Epred__set_2EIMAGE @ A_27c @ A_27b @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V0f @ V1g ) @ V2s ) ) ).
thf(thm_2Epred__set_2EFORALL__IN__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ! [V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V2s ) )
=> ( V0P @ V3y ) )
<=> ! [V4x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4x @ V2s )
=> ( V0P @ ( V1f @ V4x ) ) ) ) ).
thf(thm_2Epred__set_2EEXISTS__IN__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1f: A_27b > A_27a,V2s: A_27b > $o] :
( ? [V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V2s ) )
& ( V0P @ V3y ) )
<=> ? [V4x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4x @ V2s )
& ( V0P @ ( V1f @ V4x ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__IFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
<=> ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ V0t ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V1s ) )
=> ( ( ( V2f @ V4x )
= ( V2f @ V5y ) )
<=> ( V4x = V5y ) ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__ID,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( c_2Epred__set_2EINJ @ A_27a @ A_27a
@ ^ [V1x: A_27a] : V1x
@ V0s
@ V0s ) ).
thf(thm_2Epred__set_2EINJ__COMPOSE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2s: A_27a > $o,V3t: A_27b > $o,V4u: A_27c > $o] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V2s @ V3t )
& ( c_2Epred__set_2EINJ @ A_27b @ A_27c @ V1g @ V3t @ V4u ) )
=> ( c_2Epred__set_2EINJ @ A_27a @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ V2s @ V4u ) ) ).
thf(thm_2Epred__set_2EINJ__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1s: A_27b > $o] : ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1s )
& ! [V2s: A_27a > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V2s @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__DELETE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
=> ! [V3e: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3e @ V0s )
=> ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V3e ) @ ( c_2Epred__set_2EDELETE @ A_27b @ V1t @ ( V2f @ V3e ) ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__INSERT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1x: A_27a,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V2s ) @ V3t )
<=> ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V2s @ V3t )
& ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V1x ) @ V3t )
& ! [V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V1x )
= ( V0f @ V4y ) ) )
=> ( V1x = V4y ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__EXTEND,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2t: A_27b > $o,V3s: A_27a > $o,V4b: A_27a > A_27b] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V4b @ V3s @ V2t )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V3s ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27b @ V0y @ V2t ) ) )
=> ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b @ V1x @ V0y @ V4b ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V3s ) @ ( c_2Epred__set_2EINSERT @ A_27b @ V0y @ V2t ) ) ) ).
thf(thm_2Epred__set_2EINJ__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o,V3s0: A_27a > $o,V4t0: A_27b > $o] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3s0 @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V2t @ V4t0 ) )
=> ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V3s0 @ V4t0 ) ) ).
thf(thm_2Epred__set_2EINJ__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
=> ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V1s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s ) ) ) ).
thf(thm_2Epred__set_2EINJ__IMAGE__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s ) @ V0t ) ) ).
thf(thm_2Epred__set_2ESURJ__ID,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( c_2Epred__set_2ESURJ @ A_27a @ A_27a
@ ^ [V1x: A_27a] : V1x
@ V0s
@ V0s ) ).
thf(thm_2Epred__set_2ESURJ__COMPOSE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2s: A_27a > $o,V3t: A_27b > $o,V4u: A_27c > $o] :
( ( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V2s @ V3t )
& ( c_2Epred__set_2ESURJ @ A_27b @ A_27c @ V1g @ V3t @ V4u ) )
=> ( c_2Epred__set_2ESURJ @ A_27a @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ V2s @ V4u ) ) ).
thf(thm_2Epred__set_2ESURJ__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1s: A_27b > $o] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1s )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
& ! [V2s: A_27a > $o] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V2s @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__SURJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
<=> ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s )
= V2t ) ) ).
thf(thm_2Epred__set_2ESURJ__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b] : ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V1f @ V0s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s ) ) ).
thf(thm_2Epred__set_2ESURJ__IMP__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
=> ? [V3g: A_27b > A_27a] : ( c_2Epred__set_2EINJ @ A_27b @ A_27a @ V3g @ V1t @ V0s ) ) ).
thf(thm_2Epred__set_2EBIJ__ID,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( c_2Epred__set_2EBIJ @ A_27a @ A_27a
@ ^ [V1x: A_27a] : V1x
@ V0s
@ V0s ) ).
thf(thm_2Epred__set_2EBIJ__IMP__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
<=> ( V1x = V2y ) ) ) ).
thf(thm_2Epred__set_2EBIJ__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1s: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1s )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
& ! [V2s: A_27a > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V2s @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EBIJ__COMPOSE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2s: A_27a > $o,V3t: A_27b > $o,V4u: A_27c > $o] :
( ( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V2s @ V3t )
& ( c_2Epred__set_2EBIJ @ A_27b @ A_27c @ V1g @ V3t @ V4u ) )
=> ( c_2Epred__set_2EBIJ @ A_27a @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ V2s @ V4u ) ) ).
thf(thm_2Epred__set_2EBIJ__DELETE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
=> ! [V3e: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3e @ V0s )
=> ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V3e ) @ ( c_2Epred__set_2EDELETE @ A_27b @ V1t @ ( V2f @ V3e ) ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__IMAGE__BIJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b] :
( ? [V2t: A_27b > $o] : ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V1f @ V0s @ V2t )
=> ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V1f @ V0s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s ) ) ) ).
thf(thm_2Epred__set_2EINJ__BIJ__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s0: A_27a > $o,V2s: A_27a > $o,V3f: A_27a > A_27b] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s0 @ V2s )
& ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V3f @ V2s @ V0t ) )
=> ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V3f @ V1s0 @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V3f @ V1s0 ) ) ) ).
thf(thm_2Epred__set_2EBIJ__SYM__IMP,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
=> ? [V3g: A_27b > A_27a] : ( c_2Epred__set_2EBIJ @ A_27b @ A_27a @ V3g @ V1t @ V0s ) ) ).
thf(thm_2Epred__set_2EBIJ__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
<=> ? [V3g: A_27b > A_27a] : ( c_2Epred__set_2EBIJ @ A_27b @ A_27a @ V3g @ V1t @ V0s ) ) ).
thf(thm_2Epred__set_2EBIJ__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27c > $o,V2u: A_27b > $o] :
( ( ? [V3f: A_27a > A_27c] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27c @ V3f @ V0s @ V1t )
& ? [V4g: A_27c > A_27b] : ( c_2Epred__set_2EBIJ @ A_27c @ A_27b @ V4g @ V1t @ V2u ) )
=> ? [V5h: A_27a > A_27b] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V5h @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2EBIJ__INV,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ? [V3g: A_27b > A_27a] :
( ( c_2Epred__set_2EBIJ @ A_27b @ A_27a @ V3g @ V2t @ V1s )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s )
=> ( ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27b @ V3g @ V0f @ V4x )
= V4x ) )
& ! [V5x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5x @ V2t )
=> ( ( c_2Ecombin_2Eo @ A_27b @ A_27b @ A_27a @ V0f @ V3g @ V5x )
= V5x ) ) ) ) ).
thf(thm_2Epred__set_2EIN__FUNSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o,V2Q: A_27b > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V1P @ V2Q ) )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1P )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2Q ) ) ) ).
thf(thm_2Epred__set_2EFUNSET__applied,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o,V2Q: A_27b > $o] :
( ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V1P @ V2Q @ V0f )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1P )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2Q ) ) ) ).
thf(thm_2Epred__set_2EIN__DFUNSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o,V2Q: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b @ V1P @ V2Q ) )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1P )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ ( V2Q @ V3x ) ) ) ) ).
thf(thm_2Epred__set_2EDFUNSET__applied,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1P: A_27a > $o,V2Q: A_27a > A_27b > $o] :
( ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b @ V1P @ V2Q @ V0f )
<=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1P )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ ( V2Q @ V3x ) ) ) ) ).
thf(thm_2Epred__set_2EFUNSET__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > A_27b,V3x: A_27a] :
( ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V2f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0s @ V1t ) )
& ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s ) )
=> ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ V1t ) ) ).
thf(thm_2Epred__set_2EUNIV__FUNSET__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
= ( c_2Epred__set_2EUNIV @ ( A_27a > A_27b ) ) ) ).
thf(thm_2Epred__set_2EFUNSET__DFUNSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a > $o,V1y: A_27b > $o] :
( ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b @ V0x @ ( c_2Ecombin_2EK @ ( A_27b > $o ) @ A_27a @ V1y ) ) ) ).
thf(thm_2Epred__set_2EEMPTY__FUNSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o] :
( ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0s )
= ( c_2Epred__set_2EUNIV @ ( A_27a > A_27b ) ) ) ).
thf(thm_2Epred__set_2EFUNSET__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b] :
( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V1f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EFUNSET__INTER,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: A_27a > $o,V1b: A_27b > $o,V2c: A_27b > $o] :
( ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0a @ ( c_2Epred__set_2EINTER @ A_27b @ V1b @ V2c ) )
= ( c_2Epred__set_2EINTER @ ( A_27a > A_27b ) @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0a @ V1b ) @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V0a @ V2c ) ) ) ).
thf(thm_2Epred__set_2ESCHROEDER__CLOSE,axiom,
! [A_27a: $tType,V0x: A_27a,V1f: A_27a > A_27a,V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2Eschroeder__close @ A_27a @ V1f @ V2s ) )
<=> ? [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Earithmetic_2EFUNPOW @ ( A_27a > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f ) @ V3n @ V2s ) ) ) ).
thf(thm_2Epred__set_2ESCHROEDER__CLOSED,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V0f @ ( c_2Epred__set_2Eschroeder__close @ A_27a @ V0f @ V1s ) ) @ ( c_2Epred__set_2Eschroeder__close @ A_27a @ V0f @ V1s ) ) ).
thf(thm_2Epred__set_2ESCHROEDER__CLOSE__SUBSET,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2Eschroeder__close @ A_27a @ V0f @ V1s ) ) ).
thf(thm_2Epred__set_2ESCHROEDER__CLOSE__SET,axiom,
! [A_27a: $tType,V0f: A_27a > A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > A_27a ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27a @ V1s @ V1s ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2Eschroeder__close @ A_27a @ V0f @ V2t ) @ V1s ) ) ).
thf(thm_2Epred__set_2ESCHROEDER__BERNSTEIN__AUTO,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s )
& ? [V2f: A_27a > A_27a] : ( c_2Epred__set_2EINJ @ A_27a @ A_27a @ V2f @ V0s @ V1t ) )
=> ? [V3g: A_27a > A_27a] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27a @ V3g @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2ESCHROEDER__BERNSTEIN,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
& ? [V3g: A_27b > A_27a] : ( c_2Epred__set_2EINJ @ A_27b @ A_27a @ V3g @ V1t @ V0s ) )
=> ? [V4h: A_27a > A_27b] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V4h @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2EBIJ__INJ__SURJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V0s @ V1t )
& ? [V3g: A_27a > A_27b] : ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V3g @ V0s @ V1t ) )
=> ? [V4h: A_27a > A_27b] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V4h @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2EBIJ__ALT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > A_27b ) @ V0f @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b @ V1s @ V2t ) )
& ! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3y @ V2t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V4x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s ) @ ( c_2Emin_2E_3D @ A_27b @ V3y @ ( V0f @ V4x ) ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIJ__INSERT__IMP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1e: A_27a,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1e @ V2s ) )
& ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s ) @ V3t ) )
=> ? [V4u: A_27b > $o] :
( ( ( c_2Epred__set_2EINSERT @ A_27b @ ( V0f @ V1e ) @ V4u )
= V3t )
& ( (~) @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V1e ) @ V4u ) )
& ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V2s @ V4u ) ) ) ).
thf(thm_2Epred__set_2EBIJ__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ( V2t
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) ) ).
thf(thm_2Epred__set_2ELINV__OPT__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2s: A_27a > $o,V3f: A_27a > A_27b] :
( ( ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b @ V3f @ V2s @ V0y )
= ( c_2Eoption_2ESOME @ A_27a @ V1x ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ V2s )
& ( ( V3f @ V1x )
= V0y ) ) ) ).
thf(thm_2Epred__set_2EINJ__LINV__OPT__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1f: A_27b > A_27a] : ( c_2Epred__set_2EINJ @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Epred__set_2ELINV__OPT @ A_27b @ A_27a @ V1f @ V0s ) @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1f @ V0s ) @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Eoption_2ESOME @ A_27b ) @ V0s ) ) ).
thf(thm_2Epred__set_2EINJ__LINV__OPT,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
=> ! [V3x: A_27a,V4y: A_27b] :
( ( ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b @ V2f @ V1s @ V4y )
= ( c_2Eoption_2ESOME @ A_27a @ V3x ) )
<=> ( ( V4y
= ( V2f @ V3x ) )
& ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
& ( c_2Ebool_2EIN @ A_27b @ V4y @ V0t ) ) ) ) ).
thf(thm_2Epred__set_2ELINV__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( ( c_2Epred__set_2ELINV @ A_27a @ A_27b @ V0f @ V1s @ ( V0f @ V3x ) )
= V3x ) ) ) ).
thf(thm_2Epred__set_2EBIJ__LINV__INV,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ! [V3x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3x @ V2t )
=> ( ( V0f @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b @ V0f @ V1s @ V3x ) )
= V3x ) ) ) ).
thf(thm_2Epred__set_2EBIJ__LINV__BIJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ( c_2Epred__set_2EBIJ @ A_27b @ A_27a @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b @ V0f @ V1s ) @ V2t @ V1s ) ) ).
thf(thm_2Epred__set_2EBIJ__IFF__INV,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
<=> ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2t ) )
& ? [V4g: A_27b > A_27a] :
( ! [V5x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5x @ V2t )
=> ( c_2Ebool_2EIN @ A_27a @ ( V4g @ V5x ) @ V1s ) )
& ! [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ V1s )
=> ( ( V4g @ ( V0f @ V6x ) )
= V6x ) )
& ! [V7x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V7x @ V2t )
=> ( ( V0f @ ( V4g @ V7x ) )
= V7x ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIJ__INSERT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1e: A_27a,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s ) @ V3t )
<=> ( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1e @ V2s ) )
& ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V1e ) @ V3t )
& ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V2s @ ( c_2Epred__set_2EDELETE @ A_27b @ V3t @ ( V0f @ V1e ) ) ) )
| ( ( c_2Ebool_2EIN @ A_27a @ V1e @ V2s )
& ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V2s @ V3t ) ) ) ) ).
thf(thm_2Epred__set_2ERINV__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ! [V3x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3x @ V2t )
=> ( ( V0f @ ( c_2Epred__set_2ERINV @ A_27a @ A_27b @ V0f @ V1s @ V3x ) )
= V3x ) ) ) ).
thf(thm_2Epred__set_2ESURJ__INJ__INV,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
=> ? [V3g: A_27b > A_27a] :
( ( c_2Epred__set_2EINJ @ A_27b @ A_27a @ V3g @ V0t @ V1s )
& ! [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V0t )
=> ( ( V2f @ ( V3g @ V4y ) )
= V4y ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__EMPTY,axiom,
! [A_27a: $tType] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EFINITE__INDUCT,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( ( V0P @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( V0P @ V1s ) )
=> ! [V2e: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2e @ V1s ) )
=> ( V0P @ ( c_2Epred__set_2EINSERT @ A_27a @ V2e @ V1s ) ) ) ) )
=> ! [V3s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V3s )
=> ( V0P @ V3s ) ) ) ).
thf(thm_2Epred__set_2EFINITE__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) ).
thf(thm_2Epred__set_2EFINITE__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) ).
thf(thm_2Epred__set_2EFINITE__REST,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2EFINITE__REST__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) ) ).
thf(thm_2Epred__set_2EFINITE__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ).
thf(thm_2Epred__set_2EINTER__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__FINITE__I,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ).
thf(thm_2Epred__set_2EPSUBSET__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V1t @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ).
thf(thm_2Epred__set_2EFINITE__DIFF,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) ) ) ).
thf(thm_2Epred__set_2EFINITE__DIFF__down,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0P @ V1Q ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1Q ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V0P ) ) ).
thf(thm_2Epred__set_2EFINITE__SING,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ESING__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ESING @ A_27a @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) ) ).
thf(thm_2Epred__set_2EIMAGE__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1f: A_27a > A_27b] : ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s ) ) ) ).
thf(thm_2Epred__set_2EFINITELY__INJECTIVE__IMAGE__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1x: A_27b] :
( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2y @ ( c_2Emin_2E_3D @ A_27b @ V1x @ ( V0f @ V2y ) ) ) ) )
=> ! [V3s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V3s ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V3s ) ) ) ).
thf(thm_2Epred__set_2EINJECTIVE__IMAGE__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
<=> ( V1x = V2y ) )
=> ! [V3s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V3s ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V3s ) ) ) ).
thf(thm_2Epred__set_2EFINITE__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) ).
thf(thm_2Epred__set_2EREL__RESTRICT__EMPTY,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ).
thf(thm_2Epred__set_2EREL__RESTRICT__SUBSET,axiom,
! [A_27a: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s1 @ V0s2 )
=> ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V2R @ V1s1 ) @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V2R @ V0s2 ) ) ) ).
thf(thm_2Epred__set_2ECARD__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= c_2Enum_2E0 ) ).
thf(thm_2Epred__set_2ECARD__INSERT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1x: A_27a] :
( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V0s ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Enum_2ESUC @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__EQ__0,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= c_2Enum_2E0 )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__DELETE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1x: A_27a] :
( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V1x ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) @ ( c_2Earithmetic_2E_2D @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__INTER__LESS__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2ECARD__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1t )
=> ( ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1t ) ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__UNION__EQN,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1t ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ V1t ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__PSUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V1t @ V0s )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Epred__set_2ECARD @ A_27a @ V1t ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__EQ__CARD,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1t )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Epred__set_2ECARD @ A_27a @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t ) )
=> ( V0s = V1t ) ) ) ).
thf(thm_2Epred__set_2ECARD__SING,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Epred__set_2ESING__IFF__CARD1,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ESING @ A_27a @ V0s )
<=> ( ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2ECARD__DIFF,axiom,
! [A_27a: $tType,V0t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0t )
=> ! [V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V0t ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V0t ) ) ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__DIFF__EQN,axiom,
! [A_27a: $tType,V0t: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V0t ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V0t ) ) ) ) ) ).
thf(thm_2Epred__set_2ELESS__CARD__DIFF,axiom,
! [A_27a: $tType,V0t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0t )
=> ! [V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Eprim__rec_2E_3C @ ( c_2Epred__set_2ECARD @ A_27a @ V0t ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V0t ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIJ__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) ) ).
thf(thm_2Epred__set_2EBIJ__FINITE__SUBSET,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) )
=> ? [V3N: tyop_2Enum_2Enum] :
! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V3N @ V4n )
=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ ( V0f @ V4n ) @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t ) )
=> ( ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V1s )
= ( c_2Epred__set_2ECARD @ A_27b @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIJ__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V1s @ V2t ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V1s )
= ( c_2Epred__set_2ECARD @ A_27b @ V2t ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIJ__CARD__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0S: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0S )
=> ! [V1t: A_27b > $o,V2f: A_27a > A_27b] :
( ( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ V0S @ V1t )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V0S )
= ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__INJ__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
<=> ( V2x = V3y ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( ( c_2Epred__set_2ECARD @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) )
= ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) ) ).
thf(thm_2Epred__set_2ECARD__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) ) ).
thf(thm_2Epred__set_2ESURJ__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ! [V2t: A_27b > $o] :
( ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ( ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27b @ V2t ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__SURJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V2f @ V1s @ V0t ) )
=> ( c_2Epred__set_2EFINITE @ A_27b @ V0t ) ) ).
thf(thm_2Epred__set_2EFINITE__SURJ__BIJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2ESURJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
& ( ( c_2Epred__set_2ECARD @ A_27b @ V0t )
= ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) )
=> ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ V1s @ V0t ) ) ).
thf(thm_2Epred__set_2EFINITE__COMPLETE__INDUCTION,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ! [V1x: A_27a > $o] :
( ! [V2y: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V2y @ V1x )
=> ( V0P @ V2y ) )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V1x )
=> ( V0P @ V1x ) ) )
=> ! [V3x: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V3x )
=> ( V0P @ V3x ) ) ) ).
thf(thm_2Epred__set_2EINJ__CARD__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1f: A_27a > A_27b,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V1f @ V2s @ V0t )
=> ( ( c_2Epred__set_2ECARD @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V2s ) )
= ( c_2Epred__set_2ECARD @ A_27a @ V2s ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
& ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ ( c_2Epred__set_2ECARD @ A_27b @ V2t ) ) ) ).
thf(thm_2Epred__set_2EPHP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V2t )
& ( c_2Eprim__rec_2E_3C @ ( c_2Epred__set_2ECARD @ A_27b @ V2t ) @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) )
=> ( (~) @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t ) ) ) ).
thf(thm_2Epred__set_2EINJ__CARD__IMAGE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o,V2f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V1s @ V0t )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Epred__set_2ECARD @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V1s ) )
= ( c_2Epred__set_2ECARD @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2EIN__COUNT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0m @ ( c_2Epred__set_2Ecount @ V1n ) )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Epred__set_2ECOUNT__applied,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Ecount @ V1n @ V0m )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Epred__set_2ECOUNT__ZERO,axiom,
( ( c_2Epred__set_2Ecount @ c_2Enum_2E0 )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Epred__set_2ECOUNT__SUC,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Ecount @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0n @ ( c_2Epred__set_2Ecount @ V0n ) ) ) ).
thf(thm_2Epred__set_2EFINITE__COUNT,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ V0n ) ) ).
thf(thm_2Epred__set_2ECARD__COUNT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ECARD @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ V0n ) )
= V0n ) ).
thf(thm_2Epred__set_2ECOUNT__11,axiom,
! [V0n2: tyop_2Enum_2Enum,V1n1: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2Ecount @ V1n1 )
= ( c_2Epred__set_2Ecount @ V0n2 ) )
<=> ( V1n1 = V0n2 ) ) ).
thf(thm_2Epred__set_2ENOT__IN__FINITE,axiom,
! [A_27a: $tType] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ? [V1x: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__INHAB,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0P ) )
=> ? [V1x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V1x @ V0P ) ) ).
thf(thm_2Epred__set_2EIMAGE__11__INFINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) )
=> ! [V3s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V3s ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V3s ) ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EIN__INFINITE__NOT__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) )
=> ? [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
& ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V2t ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__INJ__NOT__SURJ,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
<=> ? [V1f: A_27a > A_27a] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27a @ V1f @ V0s @ V0s )
& ( (~) @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27a @ V1f @ V0s @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__UNIV,axiom,
! [A_27a: $tType] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ? [V0f: A_27a > A_27a] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) )
& ? [V3y: A_27a] :
! [V4x: A_27a] :
( (~)
@ ( ( V0f @ V4x )
= V3y ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__NUM__UNIV,axiom,
(~) @ ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Epred__set_2EFINITE__PSUBSET__INFINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
<=> ! [V1t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1t )
=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s )
=> ( c_2Epred__set_2EPSUBSET @ A_27a @ V1t @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__PSUBSET__UNIV,axiom,
! [A_27a: $tType] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__DIFF__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) )
=> ( (~)
@ ( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIJ__COUNT__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
<=> ? [V1c: tyop_2Enum_2Enum > A_27a,V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ V1c @ ( c_2Epred__set_2Ecount @ V2n ) @ V0s ) ) ).
thf(thm_2Epred__set_2EFINITE__BIJ__COUNT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ? [V1f: tyop_2Enum_2Enum > A_27a,V2b: tyop_2Enum_2Enum] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2Ecount @ V2b ) @ V0s ) ) ).
thf(thm_2Epred__set_2EFINITE__ISO__NUM,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ? [V1f: tyop_2Enum_2Enum > A_27a] :
( ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) )
& ( c_2Eprim__rec_2E_3C @ V3m @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) )
=> ( ( ( V1f @ V2n )
= ( V1f @ V3m ) )
=> ( V2n = V3m ) ) )
& ( V0s
= ( c_2Epred__set_2EGSPEC @ A_27a @ tyop_2Enum_2Enum
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ A_27a @ $o @ ( V1f @ V4n ) @ ( c_2Eprim__rec_2E_3C @ V4n @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__WEAK__ENUMERATE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
<=> ? [V1f: tyop_2Enum_2Enum > A_27a,V2b: tyop_2Enum_2Enum] :
! [V3e: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3e @ V0s )
<=> ? [V4n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4n @ V2b )
& ( V3e
= ( V1f @ V4n ) ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__WF__noloops,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Erelation_2EWF @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ V1s ) )
= ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__StrongOrder__WF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Erelation_2EStrongOrder @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ V1s ) ) )
=> ( c_2Erelation_2EWF @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ V1s ) ) ) ).
thf(thm_2Epred__set_2EIN__BIGUNION,axiom,
! [A_27a: $tType,V0x: A_27a,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1sos ) )
<=> ? [V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V2s )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V1sos ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__applied,axiom,
! [A_27a: $tType,V0x: A_27a,V1sos: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ V1sos @ V0x )
<=> ? [V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ V2s )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V1sos ) ) ) ).
thf(thm_2Epred__set_2EIN__BIGUNION__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o,V1s: A_27a > $o,V2y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V2y @ ( c_2Epred__set_2EBIGUNION @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( A_27b > $o ) @ V0f @ V1s ) ) )
<=> ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
& ( c_2Ebool_2EIN @ A_27b @ V2y @ ( V0f @ V3x ) ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EBIGUNION__EQ__EMPTY,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( ( ( c_2Epred__set_2EBIGUNION @ A_27a @ V0P )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( ( V0P
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) )
| ( V0P
= ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) ) )
& ( ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( c_2Epred__set_2EBIGUNION @ A_27a @ V0P ) )
<=> ( ( V0P
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) )
| ( V0P
= ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__SING,axiom,
! [A_27a: $tType,V0x: A_27a > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0x @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) )
= V0x ) ).
thf(thm_2Epred__set_2EBIGUNION__PAIR,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V1t @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
= ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2EBIGUNION__UNION,axiom,
! [A_27a: $tType,V0s1: ( A_27a > $o ) > $o,V1s2: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EUNION @ ( A_27a > $o ) @ V0s1 @ V1s2 ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s1 ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s2 ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__BIGUNION,axiom,
! [A_27a: $tType] :
( ! [V0s: ( A_27a > $o ) > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) @ V1t )
<=> ! [V2s_27: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s_27 @ V0s )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V2s_27 @ V1t ) ) )
& ! [V3s: ( A_27a > $o ) > $o,V4t: A_27a > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27a @ V4t @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V3s ) )
<=> ! [V5s_27: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V5s_27 @ V3s )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V4t @ V5s_27 ) ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__INSERT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0s @ V1P ) )
= ( c_2Epred__set_2EUNION @ A_27a @ V0s @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1P ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__SUBSET,axiom,
! [A_27a: $tType,V0X: A_27a > $o,V1P: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1P ) @ V0X )
<=> ! [V2Y: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2Y @ V1P )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2Y @ V0X ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__IMAGE__UNIV,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a > $o,V1N: tyop_2Enum_2Enum] :
( ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V1N @ V2n )
=> ( ( V0f @ V2n )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
= ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V0f @ ( c_2Epred__set_2Ecount @ V1N ) ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIGUNION,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V0P )
& ! [V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ V0P )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0P ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIGUNION__EQ,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0P ) )
<=> ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V0P )
& ! [V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ V0P )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__BIGUNION__I,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1P: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V0x @ V1P )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V0x @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1P ) ) ) ).
thf(thm_2Epred__set_2ECARD__BIGUNION__SAME__SIZED__SETS,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1s )
& ! [V2e: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2e @ V1s )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V2e )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V2e )
= V0n ) ) )
& ! [V3e1: A_27a > $o,V4e2: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3e1 @ V1s )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4e2 @ V1s )
& ( (~) @ ( V3e1 = V4e2 ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V3e1 @ V4e2 ) ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ ( A_27a > $o ) @ V1s ) @ V0n ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__COUNT,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a > $o] :
( ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( (~) @ ( V1m = V2n ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V0f @ V1m ) @ ( V0f @ V2n ) ) )
=> ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2EDISJOINT @ A_27a @ ( V0f @ V3n ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( A_27a > $o ) @ V0f @ ( c_2Epred__set_2Ecount @ V3n ) ) ) ) ) ).
thf(thm_2Epred__set_2EFORALL__IN__BIGUNION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1s: ( A_27a > $o ) > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s ) )
=> ( V0P @ V2x ) )
<=> ! [V3t: A_27a > $o,V4x: A_27a] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V4x @ V3t ) )
=> ( V0P @ V4x ) ) ) ).
thf(thm_2Epred__set_2EIN__BIGINTER,axiom,
! [A_27a: $tType,V0x: A_27a,V1B: ( A_27a > $o ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1B ) )
<=> ! [V2P: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2P @ V1B )
=> ( c_2Ebool_2EIN @ A_27a @ V0x @ V2P ) ) ) ).
thf(thm_2Epred__set_2EBIGINTER__applied,axiom,
! [A_27a: $tType,V0x: A_27a,V1B: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ V1B @ V0x )
<=> ! [V2P: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2P @ V1B )
=> ( c_2Ebool_2EIN @ A_27a @ V0x @ V2P ) ) ) ).
thf(thm_2Epred__set_2EIN__BIGINTER__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1f: A_27b > A_27a > $o,V2s: A_27b > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( A_27a > $o ) @ V1f @ V2s ) ) )
<=> ! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3y @ V2s )
=> ( c_2Ebool_2EIN @ A_27a @ V0x @ ( V1f @ V3y ) ) ) ) ).
thf(thm_2Epred__set_2EBIGINTER__INSERT,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1B: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0P @ V1B ) )
= ( c_2Epred__set_2EINTER @ A_27a @ V0P @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1B ) ) ) ).
thf(thm_2Epred__set_2EBIGINTER__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2EBIGINTER__INTER,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0P @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V1Q @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
= ( c_2Epred__set_2EINTER @ A_27a @ V0P @ V1Q ) ) ).
thf(thm_2Epred__set_2EBIGINTER__SING,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0P @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) )
= V0P ) ).
thf(thm_2Epred__set_2ESUBSET__BIGINTER,axiom,
! [A_27a: $tType,V0X: A_27a > $o,V1P: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0X @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1P ) )
<=> ! [V2Y: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2Y @ V1P )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V0X @ V2Y ) ) ) ).
thf(thm_2Epred__set_2EDISJOINT__BIGINTER,axiom,
! [A_27a: $tType,V0X: A_27a > $o,V1Y: A_27a > $o,V2P: ( A_27a > $o ) > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1Y @ V2P )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V1Y @ V0X ) )
=> ( ( c_2Epred__set_2EDISJOINT @ A_27a @ V0X @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V2P ) )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V2P ) @ V0X ) ) ) ).
thf(thm_2Epred__set_2EBIGINTER__UNION,axiom,
! [A_27a: $tType,V0s1: ( A_27a > $o ) > $o,V1s2: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EUNION @ ( A_27a > $o ) @ V0s1 @ V1s2 ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V0s1 ) @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1s2 ) ) ) ).
thf(thm_2Epred__set_2EBIGINTER__SUBSET,axiom,
! [A_27a: $tType,V0sp: A_27a > $o,V1s: ( A_27a > $o ) > $o] :
( ( ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V0sp ) )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1s ) @ V0sp ) ) ).
thf(thm_2Epred__set_2EDIFF__BIGINTER1,axiom,
! [A_27a: $tType,V0sp: A_27a > $o,V1s: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1s ) )
= ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2u: A_27a > $o] : ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ V2u )
@ V1s ) ) ) ).
thf(thm_2Epred__set_2EDIFF__BIGINTER,axiom,
! [A_27a: $tType,V0sp: A_27a > $o,V1s: ( A_27a > $o ) > $o] :
( ( ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V0sp ) )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
=> ( ( c_2Epred__set_2EBIGINTER @ A_27a @ V1s )
= ( c_2Epred__set_2EDIFF @ A_27a @ V0sp
@ ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V3u: A_27a > $o] : ( c_2Epred__set_2EDIFF @ A_27a @ V0sp @ V3u )
@ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__BIGINTER,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ? [V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ V0P )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V0P ) ) ) ).
thf(thm_2Epred__set_2EIN__CROSS,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2x @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ V0P )
& ( c_2Ebool_2EIN @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2x ) @ V1Q ) ) ) ).
thf(thm_2Epred__set_2ECROSS__applied,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q @ V2x )
<=> ( ( c_2Ebool_2EIN @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2x ) @ V0P )
& ( c_2Ebool_2EIN @ A_27b @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2x ) @ V1Q ) ) ) ).
thf(thm_2Epred__set_2ECROSS__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: A_27a > $o] :
( ( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( ( c_2Epred__set_2ECROSS @ A_27c @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27c ) @ V0P )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2ECROSS__EMPTY__EQN,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V1s @ V0t )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
<=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V0t
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ).
thf(thm_2Epred__set_2ECROSS__INSERT__LEFT,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2x: A_27a] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V0P ) @ V1Q )
= ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1Q ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) ) ) ).
thf(thm_2Epred__set_2ECROSS__INSERT__RIGHT,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2x: A_27b] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ ( c_2Epred__set_2EINSERT @ A_27b @ V2x @ V1Q ) )
= ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ ( c_2Epred__set_2EINSERT @ A_27b @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) ) ) ).
thf(thm_2Epred__set_2EFINITE__CROSS,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1Q ) )
=> ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) ) ) ).
thf(thm_2Epred__set_2ECROSS__SINGS,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2EINSERT @ A_27b @ V1y @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__SING__CROSS,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1P: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V1P )
=> ( ( c_2Epred__set_2ECARD @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1P ) )
= ( c_2Epred__set_2ECARD @ A_27b @ V1P ) ) ) ).
thf(thm_2Epred__set_2ECARD__CROSS,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1Q ) )
=> ( ( c_2Epred__set_2ECARD @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) )
= ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) @ ( c_2Epred__set_2ECARD @ A_27b @ V1Q ) ) ) ) ).
thf(thm_2Epred__set_2ECROSS__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2P0: A_27a > $o,V3Q0: A_27b > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V2P0 @ V3Q0 ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) )
<=> ( ( V2P0
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V3Q0
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
| ( ( c_2Epred__set_2ESUBSET @ A_27a @ V2P0 @ V0P )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V3Q0 @ V1Q ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__CROSS__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0P @ V1Q ) )
<=> ( ( V0P
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V1Q
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
| ( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1Q ) ) ) ) ).
thf(thm_2Epred__set_2ECROSS__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2EUNIV @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) )
= ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__PAIR__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) )
= ( c_2Epred__set_2EUNION @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ).
thf(thm_2Epred__set_2EINJ__INL,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ebool_2EIN @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) @ V0t ) )
=> ( c_2Epred__set_2EINJ @ A_27a @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V1s @ V0t ) ) ).
thf(thm_2Epred__set_2EINJ__INR,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: ( tyop_2Esum_2Esum @ A_27b @ A_27a ) > $o,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ebool_2EIN @ ( tyop_2Esum_2Esum @ A_27b @ A_27a ) @ ( c_2Esum_2EINR @ A_27b @ A_27a @ V2x ) @ V0t ) )
=> ( c_2Epred__set_2EINJ @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27a ) @ ( c_2Esum_2EINR @ A_27b @ A_27a ) @ V1s @ V0t ) ) ).
thf(thm_2Epred__set_2EIN__COMPL,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2ECOMPL @ A_27a @ V1s ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) ) ) ).
thf(thm_2Epred__set_2ECOMPL__applied,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2ECOMPL @ A_27a @ V1s @ V0x )
<=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) ) ) ).
thf(thm_2Epred__set_2ECOMPL__COMPL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) )
= V0s ) ).
thf(thm_2Epred__set_2ECOMPL__CLAUSES,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) @ V0s )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ECOMPL__SPLITS,axiom,
! [A_27a: $tType,V0p: A_27a > $o,V1q: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0p @ V1q ) @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0p ) @ V1q ) )
= V1q ) ).
thf(thm_2Epred__set_2EINTER__UNION__COMPL,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) @ ( c_2Epred__set_2ECOMPL @ A_27a @ V1t ) ) ) ) ).
thf(thm_2Epred__set_2ECOMPL__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2ECOMPL__INTER,axiom,
! [A_27a: $tType,V0x: A_27a > $o] :
( ( ( c_2Epred__set_2EINTER @ A_27a @ V0x @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0x ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0x ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ECOMPL__UNION,axiom,
! [A_27a: $tType,V0t: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V0t ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V1s ) @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0t ) ) ) ).
thf(thm_2Epred__set_2EITSET__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1P: ( A_27a > $o ) > A_27b > $o] :
( ! [V2s: A_27a > $o,V3b: A_27b] :
( ( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( V1P @ ( c_2Epred__set_2EREST @ A_27a @ V2s ) @ ( V0f @ ( c_2Epred__set_2ECHOICE @ A_27a @ V2s ) @ V3b ) ) )
=> ( V1P @ V2s @ V3b ) )
=> ! [V4v: A_27a > $o,V5v1: A_27b] : ( V1P @ V4v @ V5v1 ) ) ).
thf(thm_2Epred__set_2EITSET__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b > A_27b,V2b: A_27b] :
( ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V1f @ V0s @ V2b )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) @ ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V2b @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V1f @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) @ ( V1f @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) @ V2b ) ) ) @ ( c_2Ebool_2EARB @ A_27b ) ) ) ).
thf(thm_2Epred__set_2EITSET__IND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1P: ( A_27a > $o ) > A_27b > $o] :
( ! [V2s: A_27a > $o,V3b: A_27b] :
( ( ( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
& ( (~)
@ ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( V1P @ ( c_2Epred__set_2EREST @ A_27a @ V2s ) @ ( V0f @ ( c_2Epred__set_2ECHOICE @ A_27a @ V2s ) @ V3b ) ) )
=> ( V1P @ V2s @ V3b ) )
=> ! [V4v: A_27a > $o,V5v1: A_27b] : ( V1P @ V4v @ V5v1 ) ) ).
thf(thm_2Epred__set_2EITSET__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b > A_27b,V2b: A_27b] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V1f @ V0s @ V2b )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V2b @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V1f @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) @ ( V1f @ ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) @ V2b ) ) ) ) ) ).
thf(thm_2Epred__set_2EITSET__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1b: A_27b] :
( ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1b )
= V1b ) ).
thf(thm_2Epred__set_2EITSET__INSERT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1f: A_27a > A_27b > A_27b,V2x: A_27a,V3b: A_27b] :
( ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V1f @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V0s ) @ V3b )
= ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V1f @ ( c_2Epred__set_2EREST @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V0s ) ) @ ( V1f @ ( c_2Epred__set_2ECHOICE @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V0s ) ) @ V3b ) ) ) ) ).
thf(thm_2Epred__set_2ECOMMUTING__ITSET__INSERT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1s: A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a,V4z: A_27b] :
( ( V0f @ V2x @ ( V0f @ V3y @ V4z ) )
= ( V0f @ V3y @ ( V0f @ V2x @ V4z ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ! [V5x: A_27a,V6b: A_27b] :
( ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V5x @ V1s ) @ V6b )
= ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V5x ) @ ( V0f @ V5x @ V6b ) ) ) ) ).
thf(thm_2Epred__set_2ECOMMUTING__ITSET__RECURSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > A_27b,V1e: A_27a,V2s: A_27a > $o,V3b: A_27b] :
( ( ! [V4x: A_27a,V5y: A_27a,V6z: A_27b] :
( ( V0f @ V4x @ ( V0f @ V5y @ V6z ) )
= ( V0f @ V5y @ ( V0f @ V4x @ V6z ) ) )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2s ) )
=> ( ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s ) @ V3b )
= ( V0f @ V1e @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1e ) @ V3b ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__THM,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= c_2Enum_2E0 )
& ! [V1e: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s ) )
= ( c_2Earithmetic_2E_2B @ ( V0f @ V1e ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1e ) ) ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__SING,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1e: A_27a] :
( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( V0f @ V1e ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__SUBSET__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V2t ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__IN__LE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2e: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V2e @ V1s ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V2e ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__DELETE,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ! [V2e: A_27a] :
( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V2e ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ A_27a @ V2e @ V1s ) @ ( c_2Earithmetic_2E_2D @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) @ ( V0f @ V2e ) ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__UNION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V2t ) )
=> ( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V2t ) ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__lower__bound,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ! [V2n: tyop_2Enum_2Enum] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Earithmetic_2E_3C_3D @ V2n @ ( V0f @ V3x ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ V2n ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__upper__bound,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ! [V2n: tyop_2Enum_2Enum] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V3x ) @ V2n ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s ) @ ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ V2n ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__SAME__IMAGE,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0P )
=> ! [V1f: A_27a > tyop_2Enum_2Enum,V2p: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2p @ V0P )
& ! [V3q: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3q @ V0P )
=> ( ( V1f @ V2p )
= ( V1f @ V3q ) ) ) )
=> ( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V1f @ V0P )
= ( c_2Earithmetic_2E_2A @ ( c_2Epred__set_2ECARD @ A_27a @ V0P ) @ ( V1f @ V2p ) ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__CONG,axiom,
! [A_27a: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2f2: A_27a > tyop_2Enum_2Enum,V3f1: A_27a > tyop_2Enum_2Enum] :
( ( ( V1s1 = V0s2 )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s2 )
=> ( ( V3f1 @ V4x )
= ( V2f2 @ V4x ) ) ) )
=> ( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V3f1 @ V1s1 )
= ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V2f2 @ V0s2 ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__ZERO,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0f @ V1s )
= c_2Enum_2E0 )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( ( V0f @ V2x )
= c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Epred__set_2EABS__DIFF__SUM__IMAGE,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Enum_2Enum,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2EABS__DIFF @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V1f @ V2s ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0g @ V2s ) )
@ ( c_2Epred__set_2ESUM__IMAGE @ A_27a
@ ^ [V3x: A_27a] : ( c_2Earithmetic_2EABS__DIFF @ ( V1f @ V3x ) @ ( V0g @ V3x ) )
@ V2s ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__MONO__LESS__EQ,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Enum_2Enum,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V3x ) @ ( V0g @ V3x ) ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V1f @ V2s ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0g @ V2s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__MONO__LESS,axiom,
! [A_27a: $tType,V0g: A_27a > tyop_2Enum_2Enum,V1f: A_27a > tyop_2Enum_2Enum,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Eprim__rec_2E_3C @ ( V1f @ V3x ) @ ( V0g @ V3x ) ) )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
=> ( c_2Earithmetic_2E_3C_3D @ ( V1f @ V4x ) @ ( V0g @ V4x ) ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V1f @ V2s ) @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V0g @ V2s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__INJ__o,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ A_27b @ V0s )
=> ! [V1g: A_27b > A_27a] :
( ( c_2Epred__set_2EINJ @ A_27b @ A_27a @ V1g @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ! [V2f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V2f @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V1g @ V0s ) )
= ( c_2Epred__set_2ESUM__IMAGE @ A_27b @ ( c_2Ecombin_2Eo @ A_27b @ tyop_2Enum_2Enum @ A_27a @ V2f @ V1g ) @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__IMAGE__PERMUTES,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ! [V1g: A_27a > A_27a] :
( ( c_2Epred__set_2EBIJ @ A_27a @ A_27a @ V1g @ V0s @ V0s )
=> ! [V2f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ tyop_2Enum_2Enum @ A_27a @ V2f @ V1g ) @ V0s )
= ( c_2Epred__set_2ESUM__IMAGE @ A_27a @ V2f @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__SET__THM,axiom,
( ( ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= c_2Enum_2E0 )
& ! [V0x: tyop_2Enum_2Enum,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
=> ( ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0x @ V1s ) )
= ( c_2Earithmetic_2E_2B @ V0x @ ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EDELETE @ tyop_2Enum_2Enum @ V1s @ V0x ) ) ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__SET__EMPTY,axiom,
( ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= c_2Enum_2E0 ) ).
thf(thm_2Epred__set_2ESUM__SET__SING,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0n @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
= V0n ) ).
thf(thm_2Epred__set_2ESUM__SET__SUBSET__LE,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1t: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1t )
& ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V0s @ V1t ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ESUM__SET @ V0s ) @ ( c_2Epred__set_2ESUM__SET @ V1t ) ) ) ).
thf(thm_2Epred__set_2ESUM__SET__IN__LE,axiom,
! [V0x: tyop_2Enum_2Enum,V1s: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ V1s ) )
=> ( c_2Earithmetic_2E_3C_3D @ V0x @ ( c_2Epred__set_2ESUM__SET @ V1s ) ) ) ).
thf(thm_2Epred__set_2ESUM__SET__DELETE,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ! [V1e: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EDELETE @ tyop_2Enum_2Enum @ V0s @ V1e ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1e @ V0s ) @ ( c_2Earithmetic_2E_2D @ ( c_2Epred__set_2ESUM__SET @ V0s ) @ V1e ) @ ( c_2Epred__set_2ESUM__SET @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2ESUM__SET__UNION,axiom,
! [V0s: tyop_2Enum_2Enum > $o,V1t: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1t ) )
=> ( ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ V0s @ V1t ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ESUM__SET @ V0s ) @ ( c_2Epred__set_2ESUM__SET @ V1t ) ) @ ( c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum @ V0s @ V1t ) ) ) ) ) ).
thf(thm_2Epred__set_2EPROD__IMAGE__THM,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EPROD__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V1e: A_27a,V2s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V2s )
=> ( ( c_2Epred__set_2EPROD__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s ) )
= ( c_2Earithmetic_2E_2A @ ( V0f @ V1e ) @ ( c_2Epred__set_2EPROD__IMAGE @ A_27a @ V0f @ ( c_2Epred__set_2EDELETE @ A_27a @ V2s @ V1e ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EPROD__SET__THM,axiom,
( ( ( c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
& ! [V0x: tyop_2Enum_2Enum,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
=> ( ( c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0x @ V1s ) )
= ( c_2Earithmetic_2E_2A @ V0x @ ( c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EDELETE @ tyop_2Enum_2Enum @ V1s @ V0x ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EPROD__SET__EMPTY,axiom,
( ( c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Epred__set_2EPROD__SET__IMAGE__REDUCTION,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2x: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Enum_2Enum @ V0f @ V1s ) )
& ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( V0f @ V2x ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Enum_2Enum @ V0f @ V1s ) ) ) )
=> ( ( c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Enum_2Enum @ V0f @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) ) )
= ( c_2Earithmetic_2E_2A @ ( V0f @ V2x ) @ ( c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EIMAGE @ A_27a @ tyop_2Enum_2Enum @ V0f @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EMAX__SET__THM,axiom,
( ( ( c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= c_2Enum_2E0 )
& ! [V0e: tyop_2Enum_2Enum,V1s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1s )
=> ( ( c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0e @ V1s ) )
= ( c_2Earithmetic_2EMAX @ V0e @ ( c_2Epred__set_2EMAX__SET @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EMAX__SET__REWRITES,axiom,
! [V0e: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
= c_2Enum_2E0 )
& ( ( c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0e @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
= V0e ) ) ).
thf(thm_2Epred__set_2EMAX__SET__ELIM,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1Q: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0P )
& ( ( V0P
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
=> ( V1Q @ c_2Enum_2E0 ) )
& ! [V2x: tyop_2Enum_2Enum] :
( ( ! [V3y: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3y @ V0P )
=> ( c_2Earithmetic_2E_3C_3D @ V3y @ V2x ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2x @ V0P ) )
=> ( V1Q @ V2x ) ) )
=> ( V1Q @ ( c_2Epred__set_2EMAX__SET @ V0P ) ) ) ).
thf(thm_2Epred__set_2EMIN__SET__ELIM,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1Q: tyop_2Enum_2Enum > $o] :
( ( ( (~)
@ ( V0P
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ! [V2x: tyop_2Enum_2Enum] :
( ( ! [V3y: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3y @ V0P )
=> ( c_2Earithmetic_2E_3C_3D @ V2x @ V3y ) )
& ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2x @ V0P ) )
=> ( V1Q @ V2x ) ) )
=> ( V1Q @ ( c_2Epred__set_2EMIN__SET @ V0P ) ) ) ).
thf(thm_2Epred__set_2EMIN__SET__THM,axiom,
( ! [V0e: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EMIN__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0e @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
= V0e )
& ! [V1s: tyop_2Enum_2Enum > $o,V2e1: tyop_2Enum_2Enum,V3e2: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2EMIN__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V2e1 @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V3e2 @ V1s ) ) )
= ( c_2Earithmetic_2EMIN @ V2e1 @ ( c_2Epred__set_2EMIN__SET @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V3e2 @ V1s ) ) ) ) ) ).
thf(thm_2Epred__set_2EMIN__SET__LEM,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
=> ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EMIN__SET @ V0s ) @ V0s )
& ! [V1x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2EMIN__SET @ V0s ) @ V1x ) ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__MIN__SET,axiom,
! [A_27a: $tType,V0I: tyop_2Enum_2Enum > $o,V1J: tyop_2Enum_2Enum > $o,V2n: A_27a] :
( ( ( (~)
@ ( V0I
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ( (~)
@ ( V1J
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V0I @ V1J ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2EMIN__SET @ V1J ) @ ( c_2Epred__set_2EMIN__SET @ V0I ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__MAX__SET,axiom,
! [V0I: tyop_2Enum_2Enum > $o,V1J: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0I )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1J )
& ( c_2Epred__set_2ESUBSET @ tyop_2Enum_2Enum @ V0I @ V1J ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2EMAX__SET @ V0I ) @ ( c_2Epred__set_2EMAX__SET @ V1J ) ) ) ).
thf(thm_2Epred__set_2EMIN__SET__LEQ__MAX__SET,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2EMIN__SET @ V0s ) @ ( c_2Epred__set_2EMAX__SET @ V0s ) ) ) ).
thf(thm_2Epred__set_2EMIN__SET__UNION,axiom,
! [V0A: tyop_2Enum_2Enum > $o,V1B: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0A )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1B )
& ( (~)
@ ( V0A
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) )
& ( (~)
@ ( V1B
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) )
=> ( ( c_2Epred__set_2EMIN__SET @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ V0A @ V1B ) )
= ( c_2Earithmetic_2EMIN @ ( c_2Epred__set_2EMIN__SET @ V0A ) @ ( c_2Epred__set_2EMIN__SET @ V1B ) ) ) ) ).
thf(thm_2Epred__set_2EMAX__SET__UNION,axiom,
! [V0A: tyop_2Enum_2Enum > $o,V1B: tyop_2Enum_2Enum > $o] :
( ( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0A )
& ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V1B ) )
=> ( ( c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ V0A @ V1B ) )
= ( c_2Earithmetic_2EMAX @ ( c_2Epred__set_2EMAX__SET @ V0A ) @ ( c_2Epred__set_2EMAX__SET @ V1B ) ) ) ) ).
thf(thm_2Epred__set_2EIN__POW,axiom,
! [A_27a: $tType,V0set: A_27a > $o,V1e: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1e @ ( c_2Epred__set_2EPOW @ A_27a @ V0set ) )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V1e @ V0set ) ) ).
thf(thm_2Epred__set_2EUNIV__FUN__TO__BOOL,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EUNIV @ ( A_27a > $o ) )
= ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__POW,axiom,
! [A_27a: $tType,V0s1: A_27a > $o,V1s2: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s1 @ V1s2 )
=> ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s1 ) @ ( c_2Epred__set_2EPOW @ A_27a @ V1s2 ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__INSERT__RIGHT,axiom,
! [A_27a: $tType,V0e: A_27a,V1s1: A_27a > $o,V2s2: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s1 @ V2s2 )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1s1 @ ( c_2Epred__set_2EINSERT @ A_27a @ V0e @ V2s2 ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__DELETE__BOTH,axiom,
! [A_27a: $tType,V0s1: A_27a > $o,V1s2: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s1 @ V1s2 )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s1 @ V2x ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s2 @ V2x ) ) ) ).
thf(thm_2Epred__set_2EPOW__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( (~)
@ ( ( c_2Epred__set_2EPOW @ A_27a @ V0s )
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) ).
thf(thm_2Epred__set_2EPOW__INSERT,axiom,
! [A_27a: $tType,V0e: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0e @ V1s ) )
= ( c_2Epred__set_2EUNION @ ( A_27a > $o ) @ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V0e ) @ ( c_2Epred__set_2EPOW @ A_27a @ V1s ) ) @ ( c_2Epred__set_2EPOW @ A_27a @ V1s ) ) ) ).
thf(thm_2Epred__set_2EPOW__EQNS,axiom,
! [A_27a: $tType] :
( ( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) )
& ! [V0e: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EPOW @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0e @ V1s ) )
= ( c_2Ebool_2ELET @ ( ( A_27a > $o ) > $o ) @ ( ( A_27a > $o ) > $o )
@ ^ [V2ps: ( A_27a > $o ) > $o] : ( c_2Epred__set_2EUNION @ ( A_27a > $o ) @ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V0e ) @ V2ps ) @ V2ps )
@ ( c_2Epred__set_2EPOW @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__POW,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2ECARD__POW,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Epred__set_2ECARD @ ( A_27a > $o ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) )
= ( c_2Earithmetic_2EEXP @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__F,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V0x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V0x @ c_2Ebool_2EF ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EGSPEC__T,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V0x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V0x @ c_2Ebool_2ET ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2EGSPEC__ID,axiom,
! [A_27a: $tType,V0y: A_27a > $o] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V1x @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0y ) ) )
= V0y ) ).
thf(thm_2Epred__set_2EGSPEC__EQ,axiom,
! [A_27a: $tType,V0y: A_27a] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V1x @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V0y ) ) )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__EQ2,axiom,
! [A_27a: $tType,V0y: A_27a] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V1x @ ( c_2Emin_2E_3D @ A_27a @ V0y @ V1x ) ) )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__F__COND,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ $o )] :
( ! [V1x: A_27a] : ( (~) @ ( c_2Epair_2ESND @ A_27b @ $o @ ( V0f @ V1x ) ) )
=> ( ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a @ V0f )
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__AND,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_2F_5C @ ( V0P @ V2x ) @ ( V1Q @ V2x ) ) ) )
= ( c_2Epred__set_2EINTER @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( V0P @ V3x ) ) )
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( V1Q @ V4x ) ) ) ) ) ).
thf(thm_2Epred__set_2EGSPEC__OR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2x @ ( c_2Ebool_2E_5C_2F @ ( V0P @ V2x ) @ ( V1Q @ V2x ) ) ) )
= ( c_2Epred__set_2EUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3x @ ( V0P @ V3x ) ) )
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( V1Q @ V4x ) ) ) ) ) ).
thf(thm_2Epred__set_2EBIGUNION__partition,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2Eequiv__on @ A_27a @ V1R @ V0s )
=> ( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2Epartition @ A_27a @ V1R @ V0s ) )
= V0s ) ) ).
thf(thm_2Epred__set_2EEMPTY__NOT__IN__partition,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2Eequiv__on @ A_27a @ V1R @ V0s )
=> ( (~) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( c_2Epred__set_2Epartition @ A_27a @ V1R @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2Epartition__elements__disjoint,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2Eequiv__on @ A_27a @ V1R @ V0s )
=> ! [V2t1: A_27a > $o,V3t2: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t1 @ ( c_2Epred__set_2Epartition @ A_27a @ V1R @ V0s ) )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3t2 @ ( c_2Epred__set_2Epartition @ A_27a @ V1R @ V0s ) )
& ( (~) @ ( V2t1 = V3t2 ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V2t1 @ V3t2 ) ) ) ).
thf(thm_2Epred__set_2Epartition__elements__interrelate,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2Eequiv__on @ A_27a @ V1R @ V0s )
=> ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Epred__set_2Epartition @ A_27a @ V1R @ V0s ) )
=> ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2t )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2t ) )
=> ( V1R @ V3x @ V4y ) ) ) ) ).
thf(thm_2Epred__set_2Epartition__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Epred__set_2Epartition @ A_27a @ V0R @ V1s ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V1s ) ) ).
thf(thm_2Epred__set_2EFINITE__partition,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ ( c_2Epred__set_2Epartition @ A_27a @ V0R @ V1s ) )
& ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ ( c_2Epred__set_2Epartition @ A_27a @ V0R @ V1s ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V2t ) ) ) ) ).
thf(thm_2Epred__set_2Epartition__CARD,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2Eequiv__on @ A_27a @ V0R @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V1s )
= ( c_2Epred__set_2ESUM__IMAGE @ ( A_27a > $o ) @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2Epartition @ A_27a @ V0R @ V1s ) ) ) ) ).
thf(thm_2Epred__set_2Epairwise__UNION,axiom,
! [A_27a: $tType,V0s2: A_27a > $o,V1s1: A_27a > $o,V2R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2Epairwise @ A_27a @ V2R @ ( c_2Epred__set_2EUNION @ A_27a @ V1s1 @ V0s2 ) )
<=> ( ( c_2Epred__set_2Epairwise @ A_27a @ V2R @ V1s1 )
& ( c_2Epred__set_2Epairwise @ A_27a @ V2R @ V0s2 )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s1 )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V0s2 ) )
=> ( ( V2R @ V3x @ V4y )
& ( V2R @ V4y @ V3x ) ) ) ) ) ).
thf(thm_2Epred__set_2Epairwise__SUBSET,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2Epairwise @ A_27a @ V0R @ V2t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) )
=> ( c_2Epred__set_2Epairwise @ A_27a @ V0R @ V1s ) ) ).
thf(thm_2Epred__set_2EKoenigsLemma,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a] :
( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2y @ ( V0R @ V1x @ V2y ) ) ) )
=> ! [V3x: A_27a] :
( ( (~)
@ ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4y @ ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3x @ V4y ) ) ) ) )
=> ? [V5f: tyop_2Enum_2Enum > A_27a] :
( ( ( V5f @ c_2Enum_2E0 )
= V3x )
& ! [V6n: tyop_2Enum_2Enum] : ( V0R @ ( V5f @ V6n ) @ ( V5f @ ( c_2Enum_2ESUC @ V6n ) ) ) ) ) ) ).
thf(thm_2Epred__set_2EKoenigsLemma__WF,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( ! [V1x: A_27a] :
( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V2y @ ( V0R @ V1x @ V2y ) ) ) )
& ( c_2Erelation_2EWF @ A_27a @ ( c_2Erelation_2Einv @ A_27a @ A_27a @ V0R ) ) )
=> ! [V3x: A_27a] :
( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4y: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4y @ ( c_2Erelation_2ERTC @ A_27a @ V0R @ V3x @ V4y ) ) ) ) ) ).
thf(thm_2Epred__set_2ESET__EQ__SUBSET,axiom,
! [A_27a: $tType,V0s1: A_27a > $o,V1s2: A_27a > $o] :
( ( V0s1 = V1s2 )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s1 @ V1s2 )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s2 @ V0s1 ) ) ) ).
thf(thm_2Epred__set_2EPSUBSET__EQN,axiom,
! [A_27a: $tType,V0s1: A_27a > $o,V1s2: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s1 @ V1s2 )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s1 @ V1s2 )
& ( (~) @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1s2 @ V0s1 ) ) ) ) ).
thf(thm_2Epred__set_2EPSUBSET__SUBSET__TRANS,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V2u ) )
=> ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2ESUBSET__PSUBSET__TRANS,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o,V2u: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2EPSUBSET @ A_27a @ V1t @ V2u ) )
=> ( c_2Epred__set_2EPSUBSET @ A_27a @ V0s @ V2u ) ) ).
thf(thm_2Epred__set_2ECROSS__EQNS,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: A_27a,V1s1: A_27a > $o,V2s2: A_27b > $o] :
( ( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V2s2 )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27a @ V0a @ V1s1 ) @ V2s2 )
= ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epred__set_2EIMAGE @ A_27b @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V3y: A_27b] : ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0a @ V3y )
@ V2s2 )
@ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V1s1 @ V2s2 ) ) ) ) ).
thf(thm_2Epred__set_2Ecount__EQN,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Ecount @ V0n )
= ( c_2Ebool_2ECOND @ ( tyop_2Enum_2Enum > $o ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum )
@ ( c_2Ebool_2ELET @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > $o )
@ ^ [V1p: tyop_2Enum_2Enum] : ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V1p @ ( c_2Epred__set_2Ecount @ V1p ) )
@ ( c_2Eprim__rec_2EPRE @ V0n ) ) ) ) ).
thf(thm_2Epred__set_2EUNIQUE__MEMBER__SING,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
& ! [V2y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2y @ V1s )
=> ( V0x = V2y ) ) )
<=> ( V1s
= ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2Einj__surj,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t )
=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V3f_27: A_27b > A_27a] : ( c_2Epred__set_2ESURJ @ A_27b @ A_27a @ V3f_27 @ V2t @ V1s ) ) ) ).
thf(thm_2Epred__set_2Einfinite__rest,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2Echooser__def__compute,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o] :
( ( c_2Epred__set_2Echooser @ A_27a @ V0s @ c_2Enum_2E0 )
= ( c_2Epred__set_2ECHOICE @ A_27a @ V0s ) )
& ! [V1s: A_27a > $o,V2n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Echooser @ A_27a @ V1s @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) )
= ( c_2Epred__set_2Echooser @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V1s ) @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V3s: A_27a > $o,V4n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Echooser @ A_27a @ V3s @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V4n ) ) )
= ( c_2Epred__set_2Echooser @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V3s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V4n ) ) ) ) ) ).
thf(thm_2Epred__set_2Einfinite__num__inj,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
<=> ? [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2EINJ @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ).
thf(thm_2Epred__set_2Ecountable__image__nats,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V0f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Epred__set_2Ecountable__surj,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2ESURJ @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ) ).
thf(thm_2Epred__set_2Enum__countable,axiom,
c_2Epred__set_2Ecountable @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ).
thf(thm_2Epred__set_2Esubset__countable,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) ) ).
thf(thm_2Epred__set_2Eimage__countable,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V1s )
=> ( c_2Epred__set_2Ecountable @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) ) ).
thf(thm_2Epred__set_2ECOUNTABLE__ALT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
<=> ? [V1f: tyop_2Enum_2Enum > A_27a] :
! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
=> ? [V3n: tyop_2Enum_2Enum] :
( ( V1f @ V3n )
= V2x ) ) ) ).
thf(thm_2Epred__set_2ECOUNTABLE__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
& ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V0s ) ) ).
thf(thm_2Epred__set_2Efinite__countable,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V0s ) ) ).
thf(thm_2Epred__set_2ECOUNTABLE__COUNT,axiom,
! [V0n: tyop_2Enum_2Enum] : ( c_2Epred__set_2Ecountable @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ V0n ) ) ).
thf(thm_2Epred__set_2ECOUNTABLE__NUM,axiom,
! [V0s: tyop_2Enum_2Enum > $o] : ( c_2Epred__set_2Ecountable @ tyop_2Enum_2Enum @ V0s ) ).
thf(thm_2Epred__set_2ECOUNTABLE__IMAGE__NUM,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > A_27a,V1s: tyop_2Enum_2Enum > $o] : ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V0f @ V1s ) ) ).
thf(thm_2Epred__set_2Epair__to__num__formula,axiom,
! [V0x: tyop_2Enum_2Enum,V1y: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Epair__to__num @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0x @ V1y ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2EDIV @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2B @ V0x @ V1y ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2B @ V0x @ V1y ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V1y ) ) ).
thf(thm_2Epred__set_2Epair__to__num__inv,axiom,
( ! [V0x: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Epair__to__num @ ( c_2Epred__set_2Enum__to__pair @ V0x ) )
= V0x )
& ! [V1x: tyop_2Enum_2Enum,V2y: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Enum__to__pair @ ( c_2Epred__set_2Epair__to__num @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1x @ V2y ) ) )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1x @ V2y ) ) ) ).
thf(thm_2Epred__set_2Ecross__countable,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
& ( c_2Epred__set_2Ecountable @ A_27b @ V1t ) )
=> ( c_2Epred__set_2Ecountable @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Epred__set_2Einter__countable,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
| ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) ) ) ).
thf(thm_2Epred__set_2Einj__countable,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27b @ V2t )
& ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V0f @ V1s @ V2t ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V1s ) ) ).
thf(thm_2Epred__set_2Ebigunion__countable,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V0s )
& ! [V1x: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1x @ V0s )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V1x ) ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2Eunion__countable,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
& ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) ) ) ).
thf(thm_2Epred__set_2Eunion__countable__IFF,axiom,
! [A_27a: $tType,V0t: A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V0t ) )
<=> ( ( c_2Epred__set_2Ecountable @ A_27a @ V1s )
& ( c_2Epred__set_2Ecountable @ A_27a @ V0t ) ) ) ).
thf(thm_2Epred__set_2Einj__image__countable__IFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b] :
( ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V1f @ V0s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s ) )
=> ( ( c_2Epred__set_2Ecountable @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s ) )
= ( c_2Epred__set_2Ecountable @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2Epow__no__surj,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( (~)
@ ? [V1f: A_27a > A_27a > $o] : ( c_2Epred__set_2ESURJ @ A_27a @ ( A_27a > $o ) @ V1f @ V0s @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) ) ) ).
thf(thm_2Epred__set_2Einfinite__pow__uncountable,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( (~) @ ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2Ecountable__Usum,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) )
<=> ( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
& ( c_2Epred__set_2Ecountable @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ).
thf(thm_2Epred__set_2Ecountable__EMPTY,axiom,
! [A_27a: $tType] : ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2Ecountable__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) )
= ( c_2Epred__set_2Ecountable @ A_27a @ V1s ) ) ).
thf(thm_2Epred__set_2Ecross__countable__IFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V1s @ V0t ) )
<=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V0t
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
| ( ( c_2Epred__set_2Ecountable @ A_27a @ V1s )
& ( c_2Epred__set_2Ecountable @ A_27b @ V0t ) ) ) ) ).
thf(thm_2Epred__set_2Ecountable__Uprod,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2Ecountable @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) )
<=> ( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
& ( c_2Epred__set_2Ecountable @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ).
thf(thm_2Epred__set_2EEXPLICIT__ENUMERATE__MONO,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Earithmetic_2EFUNPOW @ ( A_27a > $o ) @ ( c_2Epred__set_2EREST @ A_27a ) @ V0n @ V1s ) @ V1s ) ).
thf(thm_2Epred__set_2EEXPLICIT__ENUMERATE__NOT__EMPTY,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( (~)
@ ( ( c_2Earithmetic_2EFUNPOW @ ( A_27a > $o ) @ ( c_2Epred__set_2EREST @ A_27a ) @ V0n @ V1s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Epred__set_2EINFINITE__EXPLICIT__ENUMERATE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( c_2Epred__set_2EINJ @ tyop_2Enum_2Enum @ A_27a
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ECHOICE @ A_27a @ ( c_2Earithmetic_2EFUNPOW @ ( A_27a > $o ) @ ( c_2Epred__set_2EREST @ A_27a ) @ V1n @ V0s ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum )
@ V0s ) ) ).
thf(thm_2Epred__set_2EBIJ__NUM__COUNTABLE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ? [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V0s ) ) ).
thf(thm_2Epred__set_2EENUMERATE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ? [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s )
<=> ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2Eenumerate @ A_27a @ V0s ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ).
thf(thm_2Epred__set_2ECOUNTABLE__ALT__BIJ,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
| ( c_2Epred__set_2EBIJ @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2Eenumerate @ A_27a @ V0s ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ) ).
thf(thm_2Epred__set_2ECOUNTABLE__ENUM,axiom,
! [A_27a: $tType,V0c: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0c )
<=> ( ( V0c
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V1f: tyop_2Enum_2Enum > A_27a] :
( V0c
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__BIGUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1M: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1M ) )
= ( c_2Epred__set_2EBIGUNION @ A_27b @ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27b > $o ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f ) @ V1M ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__DIFF,axiom,
! [A_27a: $tType,V0s1: A_27a > $o,V1s2: A_27a > $o,V2s3: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s1 @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s2 @ V2s3 ) )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s1 @ V1s2 )
& ( c_2Epred__set_2EDISJOINT @ A_27a @ V0s1 @ V2s3 ) ) ) ).
thf(thm_2Epred__set_2EINTER__SUBSET__EQN,axiom,
! [A_27a: $tType,V0B: A_27a > $o,V1A: A_27a > $o] :
( ( ( ( c_2Epred__set_2EINTER @ A_27a @ V1A @ V0B )
= V1A )
<=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1A @ V0B ) )
& ( ( ( c_2Epred__set_2EINTER @ A_27a @ V1A @ V0B )
= V0B )
<=> ( c_2Epred__set_2ESUBSET @ A_27a @ V0B @ V1A ) ) ) ).
thf(thm_2Epred__set_2EPSUBSET__SING,axiom,
! [A_27a: $tType,V0s: A_27a,V1x: A_27a > $o] :
( ( c_2Epred__set_2EPSUBSET @ A_27a @ V1x @ ( c_2Epred__set_2EINSERT @ A_27a @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V1x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Epred__set_2EINTER__UNION,axiom,
! [A_27a: $tType,V0B: A_27a > $o,V1A: A_27a > $o] :
( ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1A @ V0B ) @ V1A )
= V1A )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0B @ V1A ) @ V1A )
= V1A )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V1A @ ( c_2Epred__set_2EUNION @ A_27a @ V1A @ V0B ) )
= V1A )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V1A @ ( c_2Epred__set_2EUNION @ A_27a @ V0B @ V1A ) )
= V1A ) ) ).
thf(thm_2Epred__set_2EUNION__DELETE,axiom,
! [A_27a: $tType,V0A: A_27a > $o,V1B: A_27a > $o,V2x: A_27a] :
( ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0A @ V1B ) @ V2x )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0A @ V2x ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V1B @ V2x ) ) ) ).
thf(thm_2Epred__set_2EDELETE__SUBSET__INSERT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1e: A_27a,V2s2: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V1e ) @ V2s2 )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V2s2 ) ) ) ).
thf(thm_2Epred__set_2EIN__INSERT__EXPAND,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2P: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Epred__set_2EINSERT @ A_27a @ V1y @ V2P ) )
<=> ( ( V0x = V1y )
| ( ( (~) @ ( V0x = V1y ) )
& ( c_2Ebool_2EIN @ A_27a @ V0x @ V2P ) ) ) ) ).
thf(thm_2Epred__set_2EFINITE__INTER,axiom,
! [A_27a: $tType,V0s1: A_27a > $o,V1s2: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s1 )
| ( c_2Epred__set_2EFINITE @ A_27a @ V1s2 ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V0s1 @ V1s2 ) ) ) ).
thf(thm_2Epred__set_2EINSERT__EQ__SING,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V0s )
= ( c_2Epred__set_2EINSERT @ A_27a @ V2y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( ( V1x = V2y )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EINSERT @ A_27a @ V2y @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(thm_2Epred__set_2ECARD__UNION__LE,axiom,
! [A_27a: $tType,V0t: A_27a > $o,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( c_2Epred__set_2EFINITE @ A_27a @ V0t ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V0t ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ECARD @ A_27a @ V1s ) @ ( c_2Epred__set_2ECARD @ A_27a @ V0t ) ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__SUBSET__gen,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2u: A_27a > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2u )
& ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2u ) @ V3t ) )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ V3t ) ) ).
thf(thm_2Epred__set_2ECARD__REST,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EREST @ A_27a @ V0s ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Epred__set_2ESUBSET__DIFF__EMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Epred__set_2EDIFF__INTER__SUBSET,axiom,
! [A_27a: $tType,V0r: A_27a > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0r @ V1s )
=> ( ( c_2Epred__set_2EDIFF @ A_27a @ V0r @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) )
= ( c_2Epred__set_2EDIFF @ A_27a @ V0r @ V2t ) ) ) ).
thf(thm_2Epred__set_2EUNION__DIFF__2,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EUNION @ A_27a @ V0s @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) )
= V0s ) ).
thf(thm_2Epred__set_2Ecount__add,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2E_2B @ V0n @ V1m ) )
= ( c_2Epred__set_2EUNION @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ V0n ) @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2B @ V0n ) @ ( c_2Epred__set_2Ecount @ V1m ) ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__EQ__SING,axiom,
! [A_27a: $tType,A_27b: $tType,V0z: A_27a,V1s: A_27b > $o,V2f: A_27b > A_27a] :
( ( ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a @ V2f @ V1s )
= ( c_2Epred__set_2EINSERT @ A_27a @ V0z @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
& ! [V3x: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3x @ V1s )
=> ( ( V2f @ V3x )
= V0z ) ) ) ) ).
thf(thm_2Epred__set_2Ecount__add1,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0n @ ( c_2Epred__set_2Ecount @ V0n ) ) ) ).
thf(thm_2Epred__set_2Ecompl__insert,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V0s ) )
= ( c_2Epred__set_2EDELETE @ A_27a @ ( c_2Epred__set_2ECOMPL @ A_27a @ V0s ) @ V1x ) ) ).
thf(thm_2Epred__set_2Ein__max__set,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ! [V1x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1x @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ V1x @ ( c_2Epred__set_2EMAX__SET @ V0s ) ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__ALT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s )
= ( c_2Ecombin_2Eo @ A_27a @ $o @ A_27b @ V1s @ V0f ) ) ).
thf(thm_2Epred__set_2EIN__PREIMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) )
= ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V2x ) @ V1s ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__applied,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2x: A_27a] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s @ V2x )
= ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V2x ) @ V1s ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__UNIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EUNIV @ A_27b ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__COMPL,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2ECOMPL @ A_27b @ V1s ) )
= ( c_2Epred__set_2ECOMPL @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__UNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EUNION @ A_27b @ V1s @ V2t ) )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__INTER,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EINTER @ A_27b @ V1s @ V2t ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__BIGUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: ( A_27b > $o ) > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EBIGUNION @ A_27b @ V1s ) )
= ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( A_27b > $o ) @ ( A_27a > $o ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f ) @ V1s ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__COMP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1g: A_27b > A_27c,V2s: A_27c > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27c @ V1g @ V2s ) )
= ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V1g @ V0f ) @ V2s ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__DIFF,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EDIFF @ A_27b @ V1s @ V2t ) )
= ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__I,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27a @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( A_27a > $o ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__K,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27a @ ( c_2Ecombin_2EK @ A_27a @ A_27b @ V0x ) @ V1s )
= ( c_2Ebool_2ECOND @ ( A_27b > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s ) @ ( c_2Epred__set_2EUNIV @ A_27b ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__DISJOINT,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2EDISJOINT @ A_27b @ V1s @ V2t )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ V1s @ V2t )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V2t ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__CROSS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27c ),V1a: A_27b > $o,V2b: A_27c > $o] :
( ( c_2Epred__set_2EPREIMAGE @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V0f @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1a @ V2b ) )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2EFST @ A_27b @ A_27c ) @ V0f ) @ V1a ) @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2ESND @ A_27b @ A_27c ) @ V0f ) @ V2b ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__COMPL__INTER,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1t: A_27b > $o,V2sp: A_27a > $o] :
( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2ECOMPL @ A_27b @ V1t ) ) @ V2sp )
= ( c_2Epred__set_2EDIFF @ A_27a @ V2sp @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1t ) ) ) ).
thf(thm_2Epred__set_2EPREIMAGE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) ) ).
thf(thm_2Epred__set_2EIMAGE__PREIMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27b > $o] : ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b @ V0f @ V1s ) ) @ V1s ) ).
thf(thm_2Epred__set_2EFINITE__is__measure__maximal,axiom,
! [A_27a: $tType,V0m: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ? [V2x: A_27a] : ( c_2Epred__set_2Eis__measure__maximal @ A_27a @ V0m @ V1s @ V2x ) ) ).
thf(thm_2Epred__set_2Eis__measure__maximal__SING,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2m: A_27a > tyop_2Enum_2Enum] :
( ( c_2Epred__set_2Eis__measure__maximal @ A_27a @ V2m @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0y )
<=> ( V0y = V1x ) ) ).
thf(thm_2Epred__set_2Eis__measure__maximal__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2m: A_27a > tyop_2Enum_2Enum,V3e: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
& ( c_2Eprim__rec_2E_3C @ ( V2m @ V3e ) @ ( V2m @ V0x ) ) )
=> ( ( c_2Epred__set_2Eis__measure__maximal @ A_27a @ V2m @ ( c_2Epred__set_2EINSERT @ A_27a @ V3e @ V1s ) @ V4y )
= ( c_2Epred__set_2Eis__measure__maximal @ A_27a @ V2m @ V1s @ V4y ) ) ) ).
%------------------------------------------------------------------------------