ITP001 Axioms: ITP063^7.ax
%------------------------------------------------------------------------------
% File : ITP063^7 : 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 : cardinal.ax [Gau19]
% : HL4063^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 350 ( 33 unt; 73 typ; 0 def)
% Number of atoms : 1343 ( 173 equ; 115 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 4643 ( 115 ~; 19 |; 220 &;3979 @)
% ( 85 <=>; 225 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg;3979 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 776 ( 776 >; 0 *; 0 +; 0 <<)
% Number of symbols : 73 ( 71 usr; 5 con; 0-6 aty)
% Number of variables : 1277 ( 49 ^1111 !; 50 ?;1277 :)
% ( 67 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
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_2Ecardinal_2E_2B__c,type,
c_2Ecardinal_2E_2B__c:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > $o ) ).
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_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_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $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_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_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_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_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( 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_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_2Ecardinal_2EHAS__SIZE,type,
c_2Ecardinal_2EHAS__SIZE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum > $o ) ).
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_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
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_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_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_2Ebool_2ET,type,
c_2Ebool_2ET: $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_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_2Ecardinal_2Ebijns,type,
c_2Ecardinal_2Ebijns:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) > $o ) ).
thf(c_2Ecardinal_2Ecardeq,type,
c_2Ecardinal_2Ecardeq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ecardinal_2Ecardgeq,type,
c_2Ecardinal_2Ecardgeq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ecardinal_2Ecardgt,type,
c_2Ecardinal_2Ecardgt:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ecardinal_2Ecardleq,type,
c_2Ecardinal_2Ecardleq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
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_2Ecardinal_2Elist,type,
c_2Ecardinal_2Elist:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
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_2Ecardinal_2Eset__exp,type,
c_2Ecardinal_2Eset__exp:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > $o ) > ( A_27a > $o ) > ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) > $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_2Ecardinal_2Ecardleq__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s1: A_27a > $o,V1s2: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s1 @ V1s2 )
<=> ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2EINJ @ A_27a @ A_27b @ V2f @ V0s1 @ V1s2 ) ) ).
thf(thm_2Ecardinal_2Ecardeq__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s1: A_27a > $o,V1s2: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s1 @ V1s2 )
<=> ? [V2f: A_27a > A_27b] : ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V2f @ V0s1 @ V1s2 ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ V1n )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= V1n ) ) ) ).
thf(thm_2Ecardinal_2Elist__def,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( c_2Ecardinal_2Elist @ A_27a @ V0A )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27a )
@ ^ [V1l: tyop_2Elist_2Elist @ A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ $o @ V1l
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V2e: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V2e @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1l ) ) @ ( c_2Ebool_2EIN @ A_27a @ V2e @ V0A ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2Ebijns__def,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( c_2Ecardinal_2Ebijns @ A_27a @ V0A )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) )
@ ^ [V1f: A_27a > ( tyop_2Eoption_2Eoption @ A_27a )] :
( c_2Epair_2E_2C @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ $o @ V1f
@ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2ETHE @ A_27a ) @ V1f ) @ V0A @ V0A )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V2a: A_27a] :
( c_2Emin_2E_3D @ $o @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V0A )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V3b: A_27a] : ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V1f @ V2a ) @ ( c_2Eoption_2ESOME @ A_27a @ V3b ) ) ) ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2Ecardgeq__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b @ V0s @ V1t )
= ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2Ecardgt__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardgt @ A_27a @ A_27b @ V0s @ V1t )
<=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2Eadd__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t )
= ( c_2Epred__set_2EUNION @ ( tyop_2Esum_2Esum @ A_27a @ A_27b )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ $o @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) ) )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b
@ ^ [V3y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ $o @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V3y ) @ ( c_2Ebool_2EIN @ A_27b @ V3y @ V1t ) ) ) ) ) ).
thf(thm_2Ecardinal_2Eset__exp__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0A: A_27b > $o,V1B: A_27a > $o] :
( ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ V0A @ V1B )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) )
@ ^ [V2f: A_27a > ( tyop_2Eoption_2Eoption @ A_27b )] :
( c_2Epair_2E_2C @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ $o @ V2f
@ ( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3b: A_27a] :
( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3b @ V1B )
@ ( c_2Ebool_2E_3F @ A_27b
@ ^ [V4a: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V4a @ V0A ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( V2f @ V3b ) @ ( c_2Eoption_2ESOME @ A_27b @ V4a ) ) ) ) ) )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V5b: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V5b @ V1B ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( V2f @ V5b ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__LT__LEMMA,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V2u @ V1t ) )
& ( (~) @ ( c_2Epred__set_2EFINITE @ A_27c @ V2u ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2u @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__ADD,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1s_27 @ V0s ) )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27d @ A_27c @ V3t_27 @ V2t ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ A_27b @ A_27d ) @ ( tyop_2Esum_2Esum @ A_27a @ A_27c ) @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27c @ V0s @ V2t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD2__ABSORB__LT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27c @ V2u ) )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2u @ V0s ) )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V2u @ V1t ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ V2u @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__ABSORB,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__ABSORB,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD2__ABSORB__LE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27c @ V2u ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V0s @ V2u )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) @ V2u ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__ABSORB__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL2__ABSORB__LE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27c @ V2u ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V0s @ V2u )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) @ V2u ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__ABSORB__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__BIGUNION,axiom,
! [A_27a: $tType,A_27b: $tType,V0s1: ( A_27b > $o ) > $o,V1k: A_27a > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1k ) )
& ( c_2Ecardinal_2Ecardleq @ ( A_27b > $o ) @ A_27a @ V0s1 @ V1k )
& ! [V2e: A_27b > $o] :
( ( c_2Ebool_2EIN @ ( A_27b > $o ) @ V2e @ V0s1 )
=> ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V2e @ V1k ) ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27b @ V0s1 ) @ V1k ) ) ).
thf(thm_2Ecardinal_2ESET__SUM__CARDEQ__SET,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( ( c_2Ecardinal_2Ecardeq @ A_27a @ ( tyop_2Epair_2Eprod @ $o @ A_27a ) @ V0s @ ( c_2Epred__set_2ECROSS @ $o @ A_27a @ ( c_2Epred__set_2EINSERT @ $o @ c_2Ebool_2ET @ ( c_2Epred__set_2EINSERT @ $o @ c_2Ebool_2EF @ ( c_2Epred__set_2EEMPTY @ $o ) ) ) @ V0s ) )
& ! [V1A: A_27b > $o,V2B: A_27b > $o] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27b @ V1A @ V2B )
& ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ V1A @ V0s )
& ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ V2B @ V0s ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ ( c_2Epred__set_2EUNION @ A_27b @ V1A @ V2B ) @ V0s ) ) ) ) ).
thf(thm_2Ecardinal_2ESET__SQUARED__CARDEQ__SET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V0s @ V0s ) @ V0s ) ) ).
thf(thm_2Ecardinal_2Eset__binomial2,axiom,
! [A_27a: $tType,V0B: A_27a > $o,V1A: A_27a > $o] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1A @ V0B ) @ ( c_2Epred__set_2EUNION @ A_27a @ V1A @ V0B ) )
= ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V1A @ V1A ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V1A @ V0B ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V0B @ V1A ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V0B @ V0B ) ) ) ).
thf(thm_2Ecardinal_2Ecardleq__empty,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V0x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2Ecardleq__lt__trans,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0r: A_27a > $o,V1s: A_27b > $o,V2t: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0r @ V1s )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V2t @ V1s ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2t @ V0r ) ) ) ).
thf(thm_2Ecardinal_2Ecardlt__leq__trans,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0r: A_27a > $o,V1s: A_27b > $o,V2t: A_27c > $o] :
( ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1s @ V0r ) )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c @ V1s @ V2t ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2t @ V0r ) ) ) ).
thf(thm_2Ecardinal_2Ecardlt__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27b > $o,V1t: A_27c > $o,V2u: A_27a > $o] :
( ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V1t @ V0s ) )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V2u @ V1t ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V2u @ V0s ) ) ) ).
thf(thm_2Ecardinal_2Ecardlt__lenoteq,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
<=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( (~) @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2Ecardlt__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( (~) @ ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ V0s ) ) ) ).
thf(thm_2Ecardinal_2Ecardleq__lteq,axiom,
! [A_27a: $tType,A_27b: $tType,V0s1: A_27a > $o,V1s2: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s1 @ V1s2 )
<=> ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1s2 @ V0s1 ) )
| ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s1 @ V1s2 ) ) ) ).
thf(thm_2Ecardinal_2Ecardleq__dichotomy,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
| ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2Ecountable__cardeq,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
=> ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
= ( c_2Epred__set_2Ecountable @ A_27b @ V1t ) ) ) ).
thf(thm_2Ecardinal_2Ecountable__thm,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
= ( c_2Ecardinal_2Ecardleq @ A_27a @ tyop_2Enum_2Enum @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Ecardinal_2EIMAGE__cardleq__rwt,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27c,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V1s @ V2t )
=> ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c @ V0f @ V1s ) @ V2t ) ) ).
thf(thm_2Ecardinal_2ESUBSET__CARDLEQ,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1y: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0x @ V1y )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0x @ V1y ) ) ).
thf(thm_2Ecardinal_2ECARDLEQ__CROSS__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0x1: A_27a > $o,V1x2: A_27b > $o,V2y1: A_27c > $o,V3y2: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0x1 @ V1x2 )
& ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27d @ V2y1 @ V3y2 ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c @ V0x1 @ V2y1 ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d @ V1x2 @ V3y2 ) ) ) ).
thf(thm_2Ecardinal_2EIMAGE__cardleq,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ V1s ) ).
thf(thm_2Ecardinal_2EINFINITE__UNIV__INF,axiom,
! [A_27a: $tType] : ( (~) @ ( c_2Epred__set_2EFINITE @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(thm_2Ecardinal_2ECARDLEQ__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s1: A_27a > $o,V1s2: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V1s2 )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s1 @ V1s2 ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V0s1 ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__CARDLEQ,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0t2: A_27d > $o,V1t1: A_27c > $o,V2s2: A_27b > $o,V3s1: A_27a > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V3s1 @ V2s2 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V1t1 @ V0t2 ) )
=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V3s1 @ V1t1 )
= ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27d @ V2s2 @ V0t2 ) ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__SUBSET__CARDLEQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V1s @ V0t )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V1s @ V0t ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__CROSS__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V1s @ V0t ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a @ V0t @ V1s ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__CROSS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0t2: A_27d > $o,V1t1: A_27c > $o,V2s2: A_27b > $o,V3s1: A_27a > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V3s1 @ V2s2 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V1t1 @ V0t2 ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c @ V3s1 @ V1t1 ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d @ V2s2 @ V0t2 ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__CLE__INFINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V0t ) ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V1s @ V0t ) ) ).
thf(thm_2Ecardinal_2EEMPTY__CARDLEQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0t ) ).
thf(thm_2Ecardinal_2ECARDEQ__INSERT__RWT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V1s ) ) ).
thf(thm_2Ecardinal_2Ecardeq__INSERT,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ V1s ) @ V1s )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
| ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__0,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27a > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V0x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27c ) @ V0x )
<=> ( V0x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27b > $o,V1s1: A_27a > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V1s1 @ V0s2 )
& ( c_2Epred__set_2EFINITE @ A_27a @ V1s1 ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V1s1 )
= ( c_2Epred__set_2ECARD @ A_27b @ V0s2 ) ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27b > $o,V1s1: A_27a > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V1s1 @ V0s2 )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s1 )
= ( c_2Epred__set_2EFINITE @ A_27b @ V0s2 ) ) ) ).
thf(thm_2Ecardinal_2Ecardleq__ANTISYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2Ecardleq__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V0s @ V2u ) ) ).
thf(thm_2Ecardinal_2Ecardleq__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ V0s ) ).
thf(thm_2Ecardinal_2Ecardeq__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
& ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27c @ V0s @ V2u ) ) ).
thf(thm_2Ecardinal_2Ecardeq__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
= ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2Ecardeq__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a @ V0s @ V0s ) ).
thf(thm_2Ecardinal_2ESURJECTIVE__ON__RIGHT__INVERSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b,V2t: A_27b > $o] :
( ! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3y @ V2t )
=> ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( ( V1f @ V4x )
= V3y ) ) )
<=> ? [V5g: A_27b > A_27a] :
! [V6y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V6y @ V2t )
=> ( ( c_2Ebool_2EIN @ A_27a @ ( V5g @ V6y ) @ V0s )
& ( ( V1f @ ( V5g @ V6y ) )
= V6y ) ) ) ) ).
thf(thm_2Ecardinal_2ESURJECTIVE__RIGHT__INVERSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27b > A_27a] :
( ! [V1y: A_27a] :
? [V2x: A_27b] :
( ( V0f @ V2x )
= V1y )
<=> ? [V3g: A_27a > A_27b] :
! [V4y: A_27a] :
( ( V0f @ ( V3g @ V4y ) )
= V4y ) ) ).
thf(thm_2Ecardinal_2EFINITE__IMAGE__INJ__GENERAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1A: A_27b > $o,V2s: A_27a > $o] :
( ( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1A ) )
=> ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s ) @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V5x ) @ V1A ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__IMAGE__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1A: A_27b > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1A ) )
=> ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V4x ) @ V1A ) ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__IMAGE__INJ__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) )
=> ( ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) ) ) ).
thf(thm_2Ecardinal_2EINFINITE__IMAGE__INJ,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_2Ecardinal_2EINFINITE__NONEMPTY,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__PRODUCT__DEPENDENT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1s: A_27a > $o,V2t: A_27a > A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Epred__set_2EFINITE @ A_27b @ ( V2t @ V3x ) ) ) )
=> ( c_2Epred__set_2EFINITE @ A_27c
@ ( c_2Epred__set_2EGSPEC @ A_27c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ $o )
@ ^ [V4x: A_27a,V5y: A_27b] : ( c_2Epair_2E_2C @ A_27c @ $o @ ( V0f @ V4x @ V5y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ A_27b @ V5y @ ( V2t @ V4x ) ) ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__PRODUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( 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 )
@ ^ [V2x: A_27a,V3y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ A_27b @ V3y @ V1t ) ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2ESURJECTIVE__IMAGE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1y: A_27b] :
? [V2x: A_27a] :
( ( V0f @ V2x )
= V1y )
<=> ! [V3P: A_27b > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( V3P @ ( V0f @ V4x ) ) ) ) )
= ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V5x: A_27b] : ( c_2Epair_2E_2C @ A_27b @ $o @ V5x @ ( V3P @ V5x ) ) ) ) ) ).
thf(thm_2Ecardinal_2ESURJECTIVE__ON__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1u: A_27a > $o,V2v: A_27b > $o] :
( ! [V3t: A_27b > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ V3t @ V2v )
=> ? [V4s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V4s @ V1u )
& ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V4s )
= V3t ) ) )
<=> ! [V5y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5y @ V2v )
=> ? [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ V1u )
& ( ( V0f @ V6x )
= V5y ) ) ) ) ).
thf(thm_2Ecardinal_2ESURJECTIVE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1t: A_27b > $o] :
? [V2s: A_27a > $o] :
( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s )
= V1t )
<=> ! [V3y: A_27b] :
? [V4x: A_27a] :
( ( V0f @ V4x )
= V3y ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) )
=> ? [V2f: A_27a > A_27b] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V0s ) @ V1t )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V0s )
& ( ( V2f @ V3x )
= ( V2f @ V4y ) ) )
=> ( V3x = V4y ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__IMAGE__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( ( 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_2Ecardinal_2ECARD__IMAGE__LE,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_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2f: A_27a > A_27b] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Epred__set_2ECARD @ A_27b @ V1t ) )
& ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V2f @ V0s ) @ V1t ) )
=> ( ! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3y @ V1t )
=> ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( ( V2f @ V4x )
= V3y ) ) )
<=> ! [V5x: A_27a,V6y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V6y @ V0s )
& ( ( V2f @ V5x )
= ( V2f @ V6y ) ) )
=> ( V5x = V6y ) ) ) ) ).
thf(thm_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1f: A_27a > A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V0s ) @ V0s ) )
=> ( ! [V2y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2y @ V0s )
=> ? [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
& ( ( V1f @ V3x )
= V2y ) ) )
<=> ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V0s )
& ( ( V1f @ V4x )
= ( V1f @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__BIJECTION,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) )
=> ? [V2f: A_27a > A_27b] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ V1t ) )
& ! [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V1t )
=> ? [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s )
& ( ( V2f @ V5x )
= V4y ) ) )
& ! [V6x: A_27a,V7y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V6x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V7y @ V0s )
& ( ( V2f @ V6x )
= ( V2f @ V7y ) ) )
=> ( V6x = V7y ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__BIJECTIONS,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) )
=> ? [V2f: A_27a > A_27b,V3g: A_27b > A_27a] :
( ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
=> ( ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V4x ) @ V1t )
& ( ( V3g @ ( V2f @ V4x ) )
= V4x ) ) )
& ! [V5y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V5y @ V1t )
=> ( ( c_2Ebool_2EIN @ A_27a @ ( V3g @ V5y ) @ V0s )
& ( ( V2f @ ( V3g @ V5y ) )
= V5y ) ) ) ) ) ).
thf(thm_2Ecardinal_2ESING__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1x: A_27a] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s )
= ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) ) ).
thf(thm_2Ecardinal_2EINJECTIVE__ON__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1u: A_27a > $o] :
( ! [V2s: A_27a > $o,V3t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V2s @ V1u )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ V1u )
& ( ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V3t ) ) )
=> ( V2s = V3t ) )
<=> ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V1u )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V1u )
& ( ( V0f @ V4x )
= ( V0f @ V5y ) ) )
=> ( V4x = V5y ) ) ) ).
thf(thm_2Ecardinal_2EINJECTIVE__IMAGE,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 @ V1s )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2t ) )
=> ( V1s = V2t ) )
<=> ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0f @ V3x )
= ( V0f @ V4y ) )
=> ( V3x = V4y ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__FINITE__BIGUNION,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V0s )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) )
<=> ! [V1t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1t @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2Enum__FINITE,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
<=> ? [V1a: tyop_2Enum_2Enum] :
! [V2x: tyop_2Enum_2Enum] :
( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2x @ V0s )
=> ( c_2Earithmetic_2E_3C_3D @ V2x @ V1a ) ) ) ).
thf(thm_2Ecardinal_2Enum__FINITE__AVOID,axiom,
! [V0s: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ V0s )
=> ? [V1a: tyop_2Enum_2Enum] : ( (~) @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1a @ V0s ) ) ) ).
thf(thm_2Ecardinal_2Enum__INFINITE,axiom,
(~) @ ( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__CARD,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ V1n )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= V1n ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__0,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ c_2Enum_2E0 )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__SUC,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ ( c_2Enum_2ESUC @ V1n ) )
<=> ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V2a: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2a @ V0s )
=> ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0s @ V2a ) @ V1n ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__HAS__SIZE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
= ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__CLAUSES,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V1s @ c_2Enum_2E0 )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V1s @ ( c_2Enum_2ESUC @ V0n ) )
<=> ? [V2a: A_27a,V3t: A_27a > $o] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V3t @ V0n )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V3t ) )
& ( V1s
= ( c_2Epred__set_2EINSERT @ A_27a @ V2a @ V3t ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__SUBSET__EQ,axiom,
! [A_27a: $tType,V0a: A_27a > $o,V1b: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1b )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V0a @ V1b )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0a )
= ( c_2Epred__set_2ECARD @ A_27a @ V1b ) ) )
=> ( V0a = V1b ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__INDEX,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ V1n )
=> ? [V2f: tyop_2Enum_2Enum > A_27a] :
( ! [V3m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3m @ V1n )
=> ( c_2Ebool_2EIN @ A_27a @ ( V2f @ V3m ) @ V0s ) )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
=> ( c_2Ebool_2E_3F_21 @ tyop_2Enum_2Enum
@ ^ [V5m: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Eprim__rec_2E_3C @ V5m @ V1n ) @ ( c_2Emin_2E_3D @ A_27a @ ( V2f @ V5m ) @ V4x ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__BIGUNION__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27a > A_27b > $o,V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ V2m )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
=> ( ( c_2Epred__set_2EFINITE @ A_27b @ ( V1t @ V4x ) )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27b @ ( V1t @ V4x ) ) @ V3n ) ) ) )
=> ( c_2Earithmetic_2E_3C_3D
@ ( c_2Epred__set_2ECARD @ A_27b
@ ( c_2Epred__set_2EBIGUNION @ A_27b
@ ( c_2Epred__set_2EGSPEC @ ( A_27b > $o ) @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ ( A_27b > $o ) @ $o @ ( V1t @ V5x ) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s ) ) ) ) )
@ ( c_2Earithmetic_2E_2A @ V2m @ V3n ) ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__BOOL,axiom,
c_2Ecardinal_2EHAS__SIZE @ $o @ ( c_2Epred__set_2EUNIV @ $o ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ).
thf(thm_2Ecardinal_2ECARD__BOOL,axiom,
( ( c_2Epred__set_2ECARD @ $o @ ( c_2Epred__set_2EUNIV @ $o ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__BOOL,axiom,
c_2Epred__set_2EFINITE @ $o @ ( c_2Epred__set_2EUNIV @ $o ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__CART__UNIV,axiom,
! [A_27a: $tType,V0m: tyop_2Enum_2Enum] :
( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0m )
=> ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( c_2Earithmetic_2EEXP @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__CART__UNIV,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Earithmetic_2EEXP @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__CART__UNIV,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( c_2Ecardinal_2EHAS__SIZE @ tyop_2Enum_2Enum
@ ( 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 ) ) )
@ V0n ) ).
thf(thm_2Ecardinal_2EFINITE__NUMSEG__LT,axiom,
! [V0n: tyop_2Enum_2Enum] :
( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum
@ ( 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_2Ecardinal_2EHAS__SIZE__NUMSEG__LE,axiom,
! [V0n: tyop_2Enum_2Enum] :
( c_2Ecardinal_2EHAS__SIZE @ tyop_2Enum_2Enum
@ ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1m: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1m @ ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) )
@ ( c_2Earithmetic_2E_2B @ V0n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__NUMSEG__LE,axiom,
! [V0n: tyop_2Enum_2Enum] :
( c_2Epred__set_2EFINITE @ tyop_2Enum_2Enum
@ ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V1m: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1m @ ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n ) ) ) ) ).
thf(thm_2Ecardinal_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_2EFINITE @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ELE__CASES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
| ( c_2Earithmetic_2E_3C_3D @ V1n @ V0m ) ) ).
thf(thm_2Ecardinal_2ELT__CASES,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
| ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
| ( V0m = V1n ) ) ).
thf(thm_2Ecardinal_2ELT,axiom,
( ! [V0m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ c_2Enum_2E0 )
= c_2Ebool_2EF )
& ! [V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1m @ ( c_2Enum_2ESUC @ V2n ) )
<=> ( ( V1m = V2n )
| ( c_2Eprim__rec_2E_3C @ V1m @ V2n ) ) ) ) ).
thf(thm_2Ecardinal_2ELT__LE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ V1n )
<=> ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ( (~) @ ( V0m = V1n ) ) ) ) ).
thf(thm_2Ecardinal_2EGE,axiom,
! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V1m @ V0n )
= ( c_2Earithmetic_2E_3C_3D @ V0n @ V1m ) ) ).
thf(thm_2Ecardinal_2ELE__SUC__LT,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Enum_2ESUC @ V0m ) @ V1n )
= ( c_2Eprim__rec_2E_3C @ V0m @ V1n ) ) ).
thf(thm_2Ecardinal_2EEMPTY__set__exp,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0B: A_27c > $o,V1A: A_27b > $o] :
( ( ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ V1A @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EINSERT @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ ( c_2Ecombin_2EK @ ( tyop_2Eoption_2Eoption @ A_27b ) @ A_27a @ ( c_2Eoption_2ENONE @ A_27b ) ) @ ( c_2Epred__set_2EEMPTY @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) ) ) )
& ( ( (~)
@ ( V0B
= ( c_2Epred__set_2EEMPTY @ A_27c ) ) )
=> ( ( c_2Ecardinal_2Eset__exp @ A_27c @ A_27d @ ( c_2Epred__set_2EEMPTY @ A_27d ) @ V0B )
= ( c_2Epred__set_2EEMPTY @ ( A_27c > ( tyop_2Eoption_2Eoption @ A_27d ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2EEMPTY__set__exp__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0A: A_27b > $o] : ( c_2Ecardinal_2Ecardeq @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ tyop_2Enum_2Enum @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ V0A @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecardinal_2ESING__set__exp,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27b,V1B: A_27a > $o,V2A: A_27c > $o] :
( ( ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27b @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) @ V1B )
= ( c_2Epred__set_2EINSERT @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) )
@ ^ [V3b: A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Ebool_2EIN @ A_27a @ V3b @ V1B ) @ ( c_2Eoption_2ESOME @ A_27b @ V0x ) @ ( c_2Eoption_2ENONE @ A_27b ) )
@ ( c_2Epred__set_2EEMPTY @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) ) ) )
& ( ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27c @ V2A @ ( c_2Epred__set_2EINSERT @ A_27b @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
= ( c_2Epred__set_2EGSPEC @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27c ) ) @ A_27c
@ ^ [V4a: A_27c] :
( c_2Epair_2E_2C @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27c ) ) @ $o
@ ^ [V5b: A_27b] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27c ) @ ( c_2Emin_2E_3D @ A_27b @ V5b @ V0x ) @ ( c_2Eoption_2ESOME @ A_27c @ V4a ) @ ( c_2Eoption_2ENONE @ A_27c ) )
@ ( c_2Ebool_2EIN @ A_27c @ V4a @ V2A ) ) ) ) ) ).
thf(thm_2Ecardinal_2ESING__set__exp__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27b,V1B: A_27a > $o,V2A: A_27c > $o] :
( ( c_2Ecardinal_2Ecardeq @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ tyop_2Enum_2Enum @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ ( c_2Epred__set_2EINSERT @ A_27b @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) @ V1B ) @ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ( c_2Ecardinal_2Ecardeq @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27c ) ) @ A_27c @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27c @ V2A @ ( c_2Epred__set_2EINSERT @ A_27b @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ V2A ) ) ).
thf(thm_2Ecardinal_2EPOW__TWO__set__exp,axiom,
! [A_27a: $tType,V0A: A_27a > $o] : ( c_2Ecardinal_2Ecardeq @ ( A_27a > $o ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0A ) @ ( c_2Ecardinal_2Eset__exp @ A_27a @ tyop_2Enum_2Enum @ ( c_2Epred__set_2Ecount @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0A ) ) ).
thf(thm_2Ecardinal_2Eset__exp__count,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1A: A_27a > $o] :
( c_2Ecardinal_2Ecardeq @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ecardinal_2Eset__exp @ tyop_2Enum_2Enum @ A_27a @ V1A @ ( c_2Epred__set_2Ecount @ V0n ) )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Elist_2Elist @ A_27a )
@ ^ [V2l: tyop_2Elist_2Elist @ A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ $o @ V2l
@ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ ( c_2Elist_2ELENGTH @ A_27a @ V2l ) @ V0n )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3e: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27a @ V3e @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V2l ) ) @ ( c_2Ebool_2EIN @ A_27a @ V3e @ V1A ) ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2Eset__exp__card__cong,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27b1: $tType,A_27b2: $tType,V0b2: A_27b2 > $o,V1b1: A_27b1 > $o,V2a2: A_27a2 > $o,V3a1: A_27a1 > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a1 @ A_27a2 @ V3a1 @ V2a2 )
=> ( ( c_2Ecardinal_2Ecardeq @ A_27b1 @ A_27b2 @ V1b1 @ V0b2 )
=> ( c_2Ecardinal_2Ecardeq @ ( A_27b1 > ( tyop_2Eoption_2Eoption @ A_27a1 ) ) @ ( A_27b2 > ( tyop_2Eoption_2Eoption @ A_27a2 ) ) @ ( c_2Ecardinal_2Eset__exp @ A_27b1 @ A_27a1 @ V3a1 @ V1b1 ) @ ( c_2Ecardinal_2Eset__exp @ A_27b2 @ A_27a2 @ V2a2 @ V0b2 ) ) ) ) ).
thf(thm_2Ecardinal_2Eset__exp__cardle__cong,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0d: A_27b > $o,V1c: A_27d > $o,V2b: A_27a > $o,V3a: A_27c > $o] :
( ( ( V2b
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ( V0d
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ( ( ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V3a @ V2b )
& ( c_2Ecardinal_2Ecardleq @ A_27d @ A_27b @ V1c @ V0d ) )
=> ( c_2Ecardinal_2Ecardleq @ ( A_27d > ( tyop_2Eoption_2Eoption @ A_27c ) ) @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( c_2Ecardinal_2Eset__exp @ A_27d @ A_27c @ V3a @ V1c ) @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27a @ V2b @ V0d ) ) ) ) ).
thf(thm_2Ecardinal_2Eexp__INSERT__cardeq,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1e: A_27a,V2A: A_27b > $o] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1e @ V0s ) )
=> ( c_2Ecardinal_2Ecardeq @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ ( tyop_2Epair_2Eprod @ A_27b @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) ) @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ V2A @ ( c_2Epred__set_2EINSERT @ A_27a @ V1e @ V0s ) ) @ ( c_2Epred__set_2ECROSS @ A_27b @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ V2A @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b @ V2A @ V0s ) ) ) ) ).
thf(thm_2Ecardinal_2Eexp__count__cardeq,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1A: A_27a > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V1A ) )
& ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ A_27a @ ( c_2Ecardinal_2Eset__exp @ tyop_2Enum_2Enum @ A_27a @ V1A @ ( c_2Epred__set_2Ecount @ V0n ) ) @ V1A ) ) ).
thf(thm_2Ecardinal_2EINFINITE__Unum,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0A ) )
<=> ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0A ) ) ).
thf(thm_2Ecardinal_2Ecardleq__SURJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0B: A_27b > $o,V1A: A_27a > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V1A @ V0B )
<=> ( ? [V2f: A_27b > A_27a] : ( c_2Epred__set_2ESURJ @ A_27b @ A_27a @ V2f @ V0B @ V1A )
| ( V1A
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(thm_2Ecardinal_2EINFINITE__cardleq__INSERT,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b,V1s: A_27b > $o,V2A: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V2A ) )
=> ( ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27b @ V0x @ V1s ) @ V2A )
= ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1s @ V2A ) ) ) ).
thf(thm_2Ecardinal_2Elist__EMPTY,axiom,
! [A_27a: $tType] :
( ( c_2Ecardinal_2Elist @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EINSERT @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(thm_2Ecardinal_2Elist__SING,axiom,
! [A_27a: $tType,V0e: A_27a] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Elist_2Elist @ A_27a ) @ tyop_2Enum_2Enum @ ( c_2Ecardinal_2Elist @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0e @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ).
thf(thm_2Ecardinal_2EUNIV__list,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EUNIV @ ( tyop_2Elist_2Elist @ A_27a ) )
= ( c_2Ecardinal_2Elist @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2Elist__BIGUNION__EXP,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( c_2Ecardinal_2Ecardeq @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( c_2Ecardinal_2Elist @ A_27a @ V0A )
@ ( c_2Epred__set_2EBIGUNION @ ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) )
@ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ ( ( tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) > $o )
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Ecardinal_2Eset__exp @ tyop_2Enum_2Enum @ A_27a @ V0A @ ( c_2Epred__set_2Ecount @ V1n ) )
@ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ecardinal_2EINFINITE__A__list__BIJ__A,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0A ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ ( c_2Ecardinal_2Elist @ A_27a @ V0A ) @ V0A ) ) ).
thf(thm_2Ecardinal_2Efinite__subsets__bijection,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0A ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ ( A_27a > $o ) @ V0A
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1s @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2EFINITE @ A_27a @ V1s ) @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V0A ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__IMAGE__INJ_27,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1f: A_27a > A_27b] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V0s ) )
=> ( ( ( V1f @ V2x )
= ( V1f @ V3y ) )
<=> ( V2x = V3y ) ) )
=> ( ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V1f @ V0s ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) ) ) ).
thf(thm_2Ecardinal_2Ecountable__decomposition,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ? [V1A: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EBIGUNION @ A_27a @ V1A )
= V0s )
& ! [V2a: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2a @ V1A )
=> ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V2a ) )
& ( c_2Epred__set_2Ecountable @ A_27a @ V2a ) ) ) ) ) ).
thf(thm_2Ecardinal_2Edisjoint__countable__decomposition,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ? [V1A: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EBIGUNION @ A_27a @ V1A )
= V0s )
& ! [V2a: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2a @ V1A )
=> ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V2a ) )
& ( c_2Epred__set_2Ecountable @ A_27a @ V2a ) ) )
& ! [V3a1: A_27a > $o,V4a2: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3a1 @ V1A )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4a2 @ V1A )
& ( (~) @ ( V3a1 = V4a2 ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V3a1 @ V4a2 ) ) ) ) ).
thf(thm_2Ecardinal_2Ecount__cardle,axiom,
! [A_27a: $tType,V0n: tyop_2Enum_2Enum,V1A: A_27a > $o] :
( ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2Ecount @ V0n ) @ V1A )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V1A )
=> ( c_2Earithmetic_2E_3C_3D @ V0n @ ( c_2Epred__set_2ECARD @ A_27a @ V1A ) ) ) ) ).
thf(thm_2Ecardinal_2ECANTOR,axiom,
! [A_27a: $tType,V0A: A_27a > $o] : ( (~) @ ( c_2Ecardinal_2Ecardleq @ ( A_27a > $o ) @ A_27a @ ( c_2Epred__set_2EPOW @ A_27a @ V0A ) @ V0A ) ) ).
thf(thm_2Ecardinal_2Ecardlt__cardle,axiom,
! [A_27a: $tType,A_27b: $tType,V0B: A_27b > $o,V1A: A_27a > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V0B @ V1A ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V1A @ V0B ) ) ).
thf(thm_2Ecardinal_2Eset__exp__product,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0B2: A_27a > $o,V1B1: A_27b > $o,V2A: A_27c > $o] : ( c_2Ecardinal_2Ecardeq @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27c ) ) ) ) @ ( ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27c ) ) @ ( c_2Ecardinal_2Eset__exp @ A_27a @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27c ) ) @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27c @ V2A @ V1B1 ) @ V0B2 ) @ ( c_2Ecardinal_2Eset__exp @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27c @ V2A @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a @ V1B1 @ V0B2 ) ) ) ).
thf(thm_2Ecardinal_2ECOUNT__EQ__EMPTY,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( ( c_2Epred__set_2Ecount @ V0n )
= ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
<=> ( V0n = c_2Enum_2E0 ) ) ).
thf(thm_2Ecardinal_2EPOW__EQ__X__EXP__X,axiom,
! [A_27a: $tType,V0A: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0A ) )
=> ( c_2Ecardinal_2Ecardeq @ ( A_27a > $o ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0A ) @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27a @ V0A @ V0A ) ) ) ).
thf(thm_2Ecardinal_2Ecardeq__bijns__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0B: A_27b > $o,V1A: A_27a > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V1A @ V0B )
=> ( c_2Ecardinal_2Ecardeq @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( A_27b > ( tyop_2Eoption_2Eoption @ A_27b ) ) @ ( c_2Ecardinal_2Ebijns @ A_27a @ V1A ) @ ( c_2Ecardinal_2Ebijns @ A_27b @ V0B ) ) ) ).
thf(thm_2Ecardinal_2Ebijections__cardeq,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( c_2Ecardinal_2Ecardeq @ ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) @ ( A_27a > $o ) @ ( c_2Ecardinal_2Ebijns @ A_27a @ V0s ) @ ( c_2Epred__set_2EPOW @ A_27a @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ELEFT__IMP__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ? [V2x: A_27a] : ( V0P @ V2x )
=> V1Q )
<=> ! [V3x: A_27a] :
( ( V0P @ V3x )
=> V1Q ) ) ).
thf(thm_2Ecardinal_2ELEFT__IMP__FORALL__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: $o] :
( ( ! [V2x: A_27a] : ( V0P @ V2x )
=> V1Q )
<=> ? [V3x: A_27a] :
( ( V0P @ V3x )
=> V1Q ) ) ).
thf(thm_2Ecardinal_2ERIGHT__IMP__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ( V0P
=> ? [V2x: A_27a] : ( V1Q @ V2x ) )
<=> ? [V3x: A_27a] :
( V0P
=> ( V1Q @ V3x ) ) ) ).
thf(thm_2Ecardinal_2ERIGHT__IMP__FORALL__THM,axiom,
! [A_27a: $tType,V0P: $o,V1Q: A_27a > $o] :
( ( V0P
=> ! [V2x: A_27a] : ( V1Q @ V2x ) )
<=> ! [V3x: A_27a] :
( V0P
=> ( V1Q @ V3x ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__FINITE__BIGUNIONS,axiom,
! [A_27a: $tType,V0s: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V0s )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0s ) )
<=> ! [V1t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1t @ V0s )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECONJ__EQ__IMP,axiom,
! [V0r: $o,V1p: $o,V2q: $o] :
( ( ( V1p
& V2q )
=> V0r )
<=> ( V1p
=> ( V2q
=> V0r ) ) ) ).
thf(thm_2Ecardinal_2EIMP__CONJ__ALT,axiom,
! [V0r: $o,V1p: $o,V2q: $o] :
( ( ( V1p
& V2q )
=> V0r )
<=> ( V2q
=> ( V1p
=> V0r ) ) ) ).
thf(thm_2Ecardinal_2ELT__SUC__LE,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V0m @ ( c_2Enum_2ESUC @ V1n ) )
= ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n ) ) ).
thf(thm_2Ecardinal_2EINJECTIVE__ON__LEFT__INVERSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) )
<=> ? [V4g: A_27b > A_27a] :
! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s )
=> ( ( V4g @ ( V0f @ V5x ) )
= V5x ) ) ) ).
thf(thm_2Ecardinal_2EINJECTIVE__LEFT__INVERSE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0f @ V1x )
= ( V0f @ V2y ) )
=> ( V1x = V2y ) )
<=> ? [V3g: A_27b > A_27a] :
! [V4x: A_27a] :
( ( V3g @ ( V0f @ V4x ) )
= V4x ) ) ).
thf(thm_2Ecardinal_2EINTER__ACI,axiom,
! [A_27a: $tType,V0r: A_27a > $o,V1p: A_27a > $o,V2q: A_27a > $o] :
( ( ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V2q )
= ( c_2Epred__set_2EINTER @ A_27a @ V2q @ V1p ) )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V2q ) @ V0r )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V2q ) @ V0r ) )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V2q ) @ V0r )
= ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V2q @ V1p ) @ V0r ) )
& ( ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V1p )
= V1p )
& ( ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V1p ) @ V2q )
= ( c_2Epred__set_2EINTER @ A_27a @ V1p @ V2q ) ) ) ).
thf(thm_2Ecardinal_2ECONJ__ACI,axiom,
! [V0r: $o,V1p: $o,V2q: $o] :
( ( ( V1p
& V2q )
<=> ( V2q
& V1p ) )
& ( ( V1p
& V2q
& V0r )
<=> ( V1p
& V2q
& V0r ) )
& ( ( V1p
& V2q
& V0r )
<=> ( V2q
& V1p
& V0r ) )
& ( ( V1p
& V1p )
<=> V1p )
& ( ( V1p
& V1p
& V2q )
<=> ( V1p
& V2q ) ) ) ).
thf(thm_2Ecardinal_2EUNION__ACI,axiom,
! [A_27a: $tType,V0r: A_27a > $o,V1p: A_27a > $o,V2q: A_27a > $o] :
( ( ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V2q )
= ( c_2Epred__set_2EUNION @ A_27a @ V2q @ V1p ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V2q ) @ V0r )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V2q ) @ V0r ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V2q ) @ V0r )
= ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V2q @ V1p ) @ V0r ) )
& ( ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V1p )
= V1p )
& ( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V1p ) @ V2q )
= ( c_2Epred__set_2EUNION @ A_27a @ V1p @ V2q ) ) ) ).
thf(thm_2Ecardinal_2ELT__NZ,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n )
<=> ( (~) @ ( V0n = c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecardinal_2ELE__1,axiom,
( ! [V0n: tyop_2Enum_2Enum] :
( ( (~) @ ( V0n = c_2Enum_2E0 ) )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V0n ) )
& ! [V1n: tyop_2Enum_2Enum] :
( ( (~) @ ( V1n = c_2Enum_2E0 ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V1n ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V2n )
=> ( (~) @ ( V2n = c_2Enum_2E0 ) ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V3n )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V3n ) )
& ! [V4n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V4n )
=> ( c_2Eprim__rec_2E_3C @ c_2Enum_2E0 @ V4n ) )
& ! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V5n )
=> ( (~) @ ( V5n = c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecardinal_2EOR__EXISTS__THM,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ( ? [V2x: A_27a] : ( V0P @ V2x )
| ? [V3x: A_27a] : ( V1Q @ V3x ) )
<=> ? [V4x: A_27a] :
( ( V0P @ V4x )
| ( V1Q @ V4x ) ) ) ).
thf(thm_2Ecardinal_2EBIJECTIVE__INJECTIVE__SURJECTIVE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2t ) )
& ! [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V2t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V5x ) @ V4y ) ) ) ) )
<=> ( ! [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V6x ) @ V2t ) )
& ! [V7x: A_27a,V8y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V7x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V8y @ V1s )
& ( ( V0f @ V7x )
= ( V0f @ V8y ) ) )
=> ( V7x = V8y ) )
& ! [V9y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V9y @ V2t )
=> ? [V10x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V10x @ V1s )
& ( ( V0f @ V10x )
= V9y ) ) ) ) ) ).
thf(thm_2Ecardinal_2EBIJECTIVE__INVERSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2t ) )
& ! [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V2t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V0f @ V5x ) @ V4y ) ) ) ) )
<=> ( ! [V6x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6x @ V1s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V6x ) @ V2t ) )
& ? [V7g: A_27b > A_27a] :
( ! [V8y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V8y @ V2t )
=> ( c_2Ebool_2EIN @ A_27a @ ( V7g @ V8y ) @ V1s ) )
& ! [V9y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V9y @ V2t )
=> ( ( V0f @ ( V7g @ V9y ) )
= V9y ) )
& ! [V10x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V10x @ V1s )
=> ( ( V7g @ ( V0f @ V10x ) )
= V10x ) ) ) ) ) ).
thf(thm_2Ecardinal_2Ele__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
<=> ? [V2f: A_27a > A_27b] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ V1t ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V5y @ V0s )
& ( ( V2f @ V4x )
= ( V2f @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) ).
thf(thm_2Ecardinal_2Elt__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
<=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ) ) ).
thf(thm_2Ecardinal_2Eeq__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
<=> ? [V2f: A_27a > A_27b] :
( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ( c_2Ebool_2EIN @ A_27b @ ( V2f @ V3x ) @ V1t ) )
& ! [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V1t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V5x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s ) @ ( c_2Emin_2E_3D @ A_27b @ ( V2f @ V5x ) @ V4y ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2Ege__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b @ V0s @ V1t )
= ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2Egt__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardgt @ A_27a @ A_27b @ V0s @ V1t )
<=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ELE__C,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
<=> ? [V2g: A_27b > A_27a] :
! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
=> ? [V4y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V4y @ V1t )
& ( ( V2g @ V4y )
= V3x ) ) ) ) ).
thf(thm_2Ecardinal_2EGE__C,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b @ V0s @ V1t )
<=> ? [V2f: A_27a > A_27b] :
! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V3y @ V1t )
=> ? [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V0s )
& ( V3y
= ( V2f @ V4x ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE,axiom,
! [A_27a: $tType,V0t: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0t )
= ( c_2Ecardinal_2Ecardgeq @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0t ) ) ).
thf(thm_2Ecardinal_2EEQ__C,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
<=> ? [V2R: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ! [V3x: A_27a,V4y: A_27b] :
( ( V2R @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3x @ V4y ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V0s )
& ( c_2Ebool_2EIN @ A_27b @ V4y @ V1t ) ) )
& ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ V0s )
=> ( c_2Ebool_2E_3F_21 @ A_27b
@ ^ [V6y: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27b @ V6y @ V1t ) @ ( V2R @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V5x @ V6y ) ) ) ) )
& ! [V7y: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V7y @ V1t )
=> ( c_2Ebool_2E_3F_21 @ A_27a
@ ^ [V8x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V8x @ V0s ) @ ( V2R @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V8x @ V7y ) ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ V0s ) ).
thf(thm_2Ecardinal_2ECARD__LE__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V0s @ V2u ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( (~) @ ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LET__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V2u @ V1t ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2u @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LTE__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2u @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V2u @ V1t ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2u @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__REFL,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a @ V0s @ V0s ) ).
thf(thm_2Ecardinal_2ECARD__EQ__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
= ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__IMP__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__IMP__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__RELATIONAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1s: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27b,V4y_27: A_27b] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( V0R @ V2x @ V3y )
& ( V0R @ V2x @ V4y_27 ) )
=> ( V3y = V4y_27 ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V5y: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V5y
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V6x: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V6x @ V1s ) @ ( V0R @ V6x @ V5y ) ) ) ) )
@ V1s ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__EMPTY,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__ANTISYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
<=> ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__TOTAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
| ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__LET__TOTAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
| ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LTE__TOTAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
| ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__TOTAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
| ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
| ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__NOT__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
<=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__NOT__LT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) )
<=> ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__LE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
<=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
& ( (~) @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__LT,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
<=> ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
| ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c @ V0s @ V2t )
= ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2t @ V0s ) )
<=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27d @ A_27b @ V3t_27 @ V1s_27 ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
& ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27c @ V1t @ V2u ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27c @ V0s @ V2u ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27c @ V0s @ V2t )
= ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) ) ) ).
thf(thm_2Ecardinal_2EINFINITE__CARD__LE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
<=> ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ).
thf(thm_2Ecardinal_2EFINITE__CARD__LT,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
<=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ A_27a @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ V1t )
=> ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__UNIV,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__EQ__SUBSET,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
<=> ? [V2u: A_27b > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ V2u @ V1t )
& ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V2u ) ) ) ).
thf(thm_2Ecardinal_2ECARD__INFINITE__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
=> ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
<=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__FINITE__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
= ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1t )
& ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ V0s @ V1t ) )
=> ( c_2Epred__set_2EFINITE @ A_27b @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__INFINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__FINITE__INFINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( (~) @ ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__CARD__IMP,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__CARD__IMP,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27b @ V1t )
& ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1t )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t )
<=> ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__C,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( ( c_2Epred__set_2ECARD @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( tyop_2Esum_2Esum @ A_27b @ A_27a ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27c ) ) @ ( tyop_2Esum_2Esum @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27c ) @ V0s @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27c @ V1t @ V2u ) ) @ ( c_2Ecardinal_2E_2B__c @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) @ V2u ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a @ V1t @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ V0s @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1t @ V2u ) ) @ ( c_2Epred__set_2ECROSS @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) @ V2u ) ) ).
thf(thm_2Ecardinal_2ECARD__LDISTRIB,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27c ) ) @ ( tyop_2Esum_2Esum @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) ) @ ( c_2Epred__set_2ECROSS @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27c ) @ V0s @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27c @ V1t @ V2u ) ) @ ( c_2Ecardinal_2E_2B__c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c @ V0s @ V2u ) ) ) ).
thf(thm_2Ecardinal_2ECARD__RDISTRIB,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] : ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) @ ( tyop_2Esum_2Esum @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epred__set_2ECROSS @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) @ V2u ) @ ( c_2Ecardinal_2E_2B__c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c @ V0s @ V2u ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c @ V1t @ V2u ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__ADDR,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ V0s @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__ADDL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] : ( c_2Ecardinal_2Ecardleq @ A_27b @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ V1t @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__LE__MUL__INFINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27a @ V0s @ V0s ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V0s @ V0s ) ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( c_2Epred__set_2EFINITE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__FINITE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( c_2Epred__set_2EFINITE @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__SQUARE__INFINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V0s @ V0s ) @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__DISJOINT__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ ( tyop_2Esum_2Esum @ A_27a @ A_27a ) @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27a @ V0s @ V1t ) ) ) ).
thf(thm_2Ecardinal_2EUNION__LE__ADD__C,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ ( tyop_2Esum_2Esum @ A_27a @ A_27a ) @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Ecardinal_2ECARD__SQUARE__NUM,axiom,
c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) @ tyop_2Enum_2Enum @ ( c_2Epred__set_2ECROSS @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ).
thf(thm_2Ecardinal_2ECARD__LE__SQUARE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27a @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0s @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a @ V0s @ V0s ) ) ).
thf(thm_2Ecardinal_2EIN__CARD__MUL,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2x: A_27a,V3y: A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27b @ V3y @ V1t ) ) ) ).
thf(thm_2Ecardinal_2EIN__CARD__ADD,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27b > $o,V1s: A_27a > $o] :
( ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V1s @ V0t ) )
= ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) )
& ! [V3y: A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V3y ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b @ V1s @ V0t ) )
= ( c_2Ebool_2EIN @ A_27b @ V3y @ V0t ) ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c @ V0s @ V2t ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) ) ) ).
thf(thm_2Ecardinal_2ECARD__ADD__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( c_2Ecardinal_2Ecardeq @ ( tyop_2Esum_2Esum @ A_27a @ A_27c ) @ ( tyop_2Esum_2Esum @ A_27b @ A_27d ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27c @ V0s @ V2t ) @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__MUL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c @ V0s @ V2t ) @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__ADD,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0s: A_27a > $o,V1s_27: A_27b > $o,V2t: A_27c > $o,V3t_27: A_27d > $o] :
( ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V0s @ V1s_27 )
& ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27d @ V2t @ V3t_27 ) )
=> ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ A_27a @ A_27c ) @ ( tyop_2Esum_2Esum @ A_27b @ A_27d ) @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27c @ V0s @ V2t ) @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27d @ V1s_27 @ V3t_27 ) ) ) ).
thf(thm_2Ecardinal_2Emul__c,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t )
= ( 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 )
@ ^ [V2x: A_27a,V3y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s ) @ ( c_2Ebool_2EIN @ A_27b @ V3y @ V1t ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ V1s ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__IMAGE__GEN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o,V2t: A_27b > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27b @ V2t @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) )
=> ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V2t @ V1s ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__IMAGE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] : ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) @ V1s ) ).
thf(thm_2Ecardinal_2ECARD__HAS__SIZE__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2n: tyop_2Enum_2Enum] :
( ( ( c_2Ecardinal_2EHAS__SIZE @ A_27a @ V0s @ V2n )
& ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V0s @ V1t ) )
=> ( c_2Ecardinal_2EHAS__SIZE @ A_27b @ V1t @ V2n ) ) ).
thf(thm_2Ecardinal_2ECARD__LT__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27a > $o,V1t: A_27b > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Epred__set_2EFINITE @ A_27b @ V1t ) )
=> ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a @ V1t @ V0s ) )
<=> ( c_2Eprim__rec_2E_3C @ ( c_2Epred__set_2ECARD @ A_27a @ V0s ) @ ( c_2Epred__set_2ECARD @ A_27b @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECARDLEQ__CARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27b > $o,V1s1: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s1 )
& ( c_2Epred__set_2EFINITE @ A_27b @ V0s2 ) )
=> ( ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b @ V1s1 @ V0s2 )
= ( c_2Earithmetic_2E_3C_3D @ ( c_2Epred__set_2ECARD @ A_27a @ V1s1 ) @ ( c_2Epred__set_2ECARD @ A_27b @ V0s2 ) ) ) ) ).
thf(thm_2Ecardinal_2ECARDEQ__CARD__EQN,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27b > $o,V1s1: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s1 )
& ( c_2Epred__set_2EFINITE @ A_27b @ V0s2 ) )
=> ( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V1s1 @ V0s2 )
<=> ( ( c_2Epred__set_2ECARD @ A_27a @ V1s1 )
= ( c_2Epred__set_2ECARD @ A_27b @ V0s2 ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__CARDEQ__I,axiom,
! [A_27a: $tType,A_27b: $tType,V0s2: A_27b > $o,V1s1: A_27a > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s1 )
& ( c_2Epred__set_2EFINITE @ A_27b @ V0s2 )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V1s1 )
= ( c_2Epred__set_2ECARD @ A_27b @ V0s2 ) ) )
=> ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b @ V1s1 @ V0s2 ) ) ).
thf(thm_2Ecardinal_2EBIJ__functions__agree,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: A_27a > A_27b,V2s: A_27a > $o,V3t: A_27b > $o] :
( ( ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V0f @ V2s @ V3t )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V2s )
=> ( ( V0f @ V4x )
= ( V1g @ V4x ) ) ) )
=> ( c_2Epred__set_2EBIJ @ A_27a @ A_27b @ V1g @ V2s @ V3t ) ) ).
thf(thm_2Ecardinal_2ECARD__MUL__LT__INFINITE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0s: A_27a > $o,V1t: A_27b > $o,V2u: A_27c > $o] :
( ( ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a @ V2u @ V0s ) )
& ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b @ V2u @ V1t ) )
& ( (~) @ ( c_2Epred__set_2EFINITE @ A_27c @ V2u ) ) )
=> ( (~) @ ( c_2Ecardinal_2Ecardleq @ A_27c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2u @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b @ V0s @ V1t ) ) ) ) ).
thf(thm_2Ecardinal_2ECANTOR__THM,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( (~)
@ ( c_2Ecardinal_2Ecardleq @ ( A_27a > $o ) @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1t: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1t @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1t @ V0s ) ) )
@ V0s ) ) ).
thf(thm_2Ecardinal_2ECANTOR__THM__UNIV,axiom,
! [A_27a: $tType] : ( (~) @ ( c_2Ecardinal_2Ecardleq @ ( A_27a > $o ) @ A_27a @ ( c_2Epred__set_2EUNIV @ ( A_27a > $o ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Ecardinal_2ENUM__COUNTABLE,axiom,
c_2Epred__set_2Ecountable @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__ALT__cardleq,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
= ( c_2Ecardinal_2Ecardleq @ A_27a @ tyop_2Enum_2Enum @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__CASES,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
| ( c_2Ecardinal_2Ecardeq @ A_27a @ tyop_2Enum_2Enum @ V0s @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ecardinal_2ECARD__LE__COUNTABLE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V1t )
& ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a @ V0s @ V1t ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__EQ__COUNTABLE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V1t )
& ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a @ V0s @ V1t ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V0s ) ) ).
thf(thm_2Ecardinal_2ECARD__COUNTABLE__CONG,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a @ V0s @ V1t )
=> ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
= ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__RESTRICT,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
=> ( c_2Epred__set_2Ecountable @ A_27a
@ ( 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 ) @ ( V1P @ V2x ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2EFINITE__IMP__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_2Ecardinal_2ECOUNTABLE__IMAGE,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_2Ecardinal_2ECOUNTABLE__IMAGE__INJ__GENERAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1A: A_27b > $o,V2s: A_27a > $o] :
( ( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V2s )
& ( ( V0f @ V3x )
= ( V0f @ V4y ) ) )
=> ( V3x = V4y ) )
& ( c_2Epred__set_2Ecountable @ A_27b @ V1A ) )
=> ( c_2Epred__set_2Ecountable @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V5x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V5x @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V5x @ V2s ) @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V5x ) @ V1A ) ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__IMAGE__INJ__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1s: A_27a > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s )
& ( ( V0f @ V2x )
= ( V0f @ V3y ) ) )
=> ( V2x = V3y ) )
=> ( ( c_2Epred__set_2Ecountable @ A_27b @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V1s ) )
= ( c_2Epred__set_2Ecountable @ A_27a @ V1s ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__IMAGE__INJ,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1A: A_27b > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( ( V0f @ V2x )
= ( V0f @ V3y ) )
=> ( V2x = V3y ) )
& ( c_2Epred__set_2Ecountable @ A_27b @ V1A ) )
=> ( c_2Epred__set_2Ecountable @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V4x ) @ V1A ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__EMPTY,axiom,
! [A_27a: $tType] : ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__INTER,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_2Ecardinal_2ECOUNTABLE__UNION__IMP,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_2Ecardinal_2ECOUNTABLE__UNION,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) )
<=> ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
& ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__SING,axiom,
! [A_27a: $tType,V0x: A_27a] : ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EINSERT @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Ecardinal_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_2Ecardinal_2ECOUNTABLE__DELETE,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V1s @ V0x ) )
= ( c_2Epred__set_2Ecountable @ A_27a @ V1s ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__DIFF__FINITE,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ V1t @ V0s ) )
= ( c_2Epred__set_2Ecountable @ A_27a @ V1t ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__CROSS,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_2Ecardinal_2ECOUNTABLE__AS__IMAGE__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
=> ? [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__AS__IMAGE__SUBSET__EQ,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
<=> ? [V1f: tyop_2Enum_2Enum > A_27a] : ( c_2Epred__set_2ESUBSET @ A_27a @ V0s @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__AS__IMAGE,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] :
( V0s
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ).
thf(thm_2Ecardinal_2EFORALL__COUNTABLE__AS__IMAGE,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o] :
( ! [V1d: A_27a > $o] :
( ( c_2Epred__set_2Ecountable @ A_27a @ V1d )
=> ( V0P @ V1d ) )
<=> ( ( V0P @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2f: tyop_2Enum_2Enum > A_27a] : ( V0P @ ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V2f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__AS__INJECTIVE__IMAGE,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V0s )
& ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ V0s ) ) )
=> ? [V1f: tyop_2Enum_2Enum > A_27a] :
( ( V0s
= ( c_2Epred__set_2EIMAGE @ tyop_2Enum_2Enum @ A_27a @ V1f @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) )
& ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( ( V1f @ V2m )
= ( V1f @ V3n ) )
=> ( V2m = V3n ) ) ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__BIGUNION,axiom,
! [A_27a: $tType,V0A: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V0A )
& ! [V1s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ V0A )
=> ( c_2Epred__set_2Ecountable @ A_27a @ V1s ) ) )
=> ( c_2Epred__set_2Ecountable @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V0A ) ) ) ).
thf(thm_2Ecardinal_2EIN__ELIM__PAIR__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o,V1a: A_27a,V2b: A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1a @ V2b )
@ ( 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 ) @ ( V0P @ V3x @ V4y ) ) ) ) )
= ( V0P @ V1a @ V2b ) ) ).
thf(thm_2Ecardinal_2ECOUNTABLE__PRODUCT__DEPENDENT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1s: A_27a > $o,V2t: A_27a > A_27b > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V1s )
& ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
=> ( c_2Epred__set_2Ecountable @ A_27b @ ( V2t @ V3x ) ) ) )
=> ( c_2Epred__set_2Ecountable @ A_27c
@ ( c_2Epred__set_2EGSPEC @ A_27c @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ $o )
@ ^ [V4x: A_27a,V5y: A_27b] : ( c_2Epair_2E_2C @ A_27c @ $o @ ( V0f @ V4x @ V5y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s ) @ ( c_2Ebool_2EIN @ A_27b @ V5y @ ( V2t @ V4x ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------