ITP001 Axioms: ITP063^5.ax
%------------------------------------------------------------------------------
% File : ITP063^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : cardinal^2.ax [Gau20]
% : HL4063^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 288 ( 3 unt; 9 typ; 0 def)
% Number of atoms : 6003 ( 140 equ; 0 cnn)
% Maximal formula atoms : 76 ( 20 avg)
% Number of connectives : 10657 ( 114 ~; 18 |; 266 &;9328 @)
% ( 110 <=>; 821 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 16 avg;9328 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 80 usr; 72 con; 0-2 aty)
% Number of variables : 1209 ( 49 ^1111 !; 49 ?;1209 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ecardinal_2E_2B__c,type,
c_2Ecardinal_2E_2B__c: del > del > $i ).
thf(mem_c_2Ecardinal_2E_2B__c,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Ecardinal_2EHAS__SIZE,type,
c_2Ecardinal_2EHAS__SIZE: del > $i ).
thf(mem_c_2Ecardinal_2EHAS__SIZE,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Ebijns,type,
c_2Ecardinal_2Ebijns: del > $i ).
thf(mem_c_2Ecardinal_2Ebijns,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecardinal_2Ebijns @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Ecardeq,type,
c_2Ecardinal_2Ecardeq: del > del > $i ).
thf(mem_c_2Ecardinal_2Ecardeq,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Ecardgeq,type,
c_2Ecardinal_2Ecardgeq: del > del > $i ).
thf(mem_c_2Ecardinal_2Ecardgeq,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Ecardgt,type,
c_2Ecardinal_2Ecardgt: del > del > $i ).
thf(mem_c_2Ecardinal_2Ecardgt,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2Ecardgt @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Ecardleq,type,
c_2Ecardinal_2Ecardleq: del > del > $i ).
thf(mem_c_2Ecardinal_2Ecardleq,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Elist,type,
c_2Ecardinal_2Elist: del > $i ).
thf(mem_c_2Ecardinal_2Elist,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecardinal_2Elist @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Ecardinal_2Eset__exp,type,
c_2Ecardinal_2Eset__exp: del > del > $i ).
thf(mem_c_2Ecardinal_2Eset__exp,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ bool ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Ecardeq__def,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ V0s1 ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardeq__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardeq__SYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardeq__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27c ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Ecardleq__def,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V2f ) @ V0s1 ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__ANTISYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__CARD,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__0,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V0x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27c ) ) @ V0x ) )
<=> ( V0x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardeq__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
| ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__INSERT__RWT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EEMPTY__CARDLEQ,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0t ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__CLE__INFINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__CROSS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( arr @ A_27c @ bool ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t1 ) @ V3t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c ) @ V0s1 ) @ V2t1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d ) @ V1s2 ) @ V3t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__CROSS__SYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__SUBSET__CARDLEQ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__CARDLEQ,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( arr @ A_27c @ bool ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t1 ) @ V3t2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V0s1 ) @ V2t1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27d ) @ V1s2 ) @ V3t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDLEQ__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__UNIV__INF,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EIMAGE__cardleq,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDLEQ__CROSS__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0x1: $i] :
( ( mem @ V0x1 @ ( arr @ A_27a @ bool ) )
=> ! [V1x2: $i] :
( ( mem @ V1x2 @ ( arr @ A_27b @ bool ) )
=> ! [V2y1: $i] :
( ( mem @ V2y1 @ ( arr @ A_27c @ bool ) )
=> ! [V3y2: $i] :
( ( mem @ V3y2 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0x1 ) @ V1x2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27d ) @ V2y1 ) @ V3y2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c ) @ V0x1 ) @ V2y1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d ) @ V1x2 ) @ V3y2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESUBSET__CARDLEQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0x ) @ V1y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EIMAGE__cardleq__rwt,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c ) @ V0f ) @ V1s ) ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecountable__thm,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ ty_2Enum_2Enum ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecountable__cardeq,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__dichotomy,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__lteq,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1s2 ) @ V0s1 ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardlt__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardlt__lenoteq,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardlt__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27c @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V1t ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V2u ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V2u ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardlt__leq__trans,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1s ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c ) @ V1s ) @ V2t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2t ) @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__lt__trans,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V2t ) @ V1s ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2t ) @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__empty,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V0x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eset__binomial2,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0A ) @ V1B ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0A ) @ V1B ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V0A ) @ V0A ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V0A ) @ V1B ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V1B ) @ V0A ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V1B ) @ V1B ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESET__SQUARED__CARDEQ__SET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESET__SUM__CARDEQ__SET,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ ( ty_2Epair_2Eprod @ bool @ A_27a ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ bool @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ bool ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ bool ) @ c_2Ebool_2EF ) @ ( c_2Epred__set_2EEMPTY @ bool ) ) ) ) @ V0s ) ) )
& ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27b @ bool ) )
=> ! [V2B: $i] :
( ( mem @ V2B @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ V1A ) @ V2B ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ V1A ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ V2B ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V1A ) @ V2B ) ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__BIGUNION,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ ( arr @ A_27a @ bool ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0k ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( arr @ A_27b @ bool ) @ A_27a ) @ V1s1 ) @ V0k ) )
& ! [V2e: $i] :
( ( mem @ V2e @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27b @ bool ) ) @ V2e ) @ V1s1 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V2e ) @ V0k ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ V1s1 ) ) @ V0k ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Eset__exp__def,axiom,
! [A_27a: del,A_27b: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27b @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ V0A ) @ V1B )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ^ [V2f: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ bool ) @ V2f )
@ ( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3b: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3b ) @ V1B ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V4a: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4a ) @ V0A ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ V2f @ V3b ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V4a ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5b: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5b ) @ V1B ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ V2f @ V5b ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EBIJ__functions__agree,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V3t ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
=> ( ( ap @ V0f @ V4x )
= ( ap @ V1g @ V4x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V1g ) @ V2s ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__CARDEQ__I,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1s2 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDEQ__CARD__EQN,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARDLEQ__CARD,axiom,
! [A_27a: del,A_27b: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s1 ) @ V1s2 ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s1 ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EEMPTY__set__exp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27b @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27c @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ V0A ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) @ ( ap @ ( c_2Ecombin_2EK @ ( ty_2Eoption_2Eoption @ A_27b ) @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ) )
& ( ( V1B
!= ( c_2Epred__set_2EEMPTY @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27c @ A_27d ) @ ( c_2Epred__set_2EEMPTY @ A_27d ) ) @ V1B )
= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ A_27d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EEMPTY__set__exp__CARD,axiom,
! [A_27a: del,A_27b: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ V0A ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESING__set__exp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ! [V2A: $i] :
( ( mem @ V2A @ ( arr @ A_27c @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ V1B )
= ( ap
@ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
@ ( lam @ A_27a
@ ^ [V3b: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3b ) @ V1B ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V0x ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) )
@ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ) )
& ( ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27c ) @ V2A ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ A_27c )
@ ( lam @ A_27c
@ ^ [V4a: $i] :
( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ bool )
@ ( lam @ A_27b
@ ^ [V5b: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V5b ) @ V0x ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27c ) @ V4a ) ) @ ( c_2Eoption_2ENONE @ A_27c ) ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27c ) @ V4a ) @ V2A ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESING__set__exp__CARD,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ! [V2A: $i] :
( ( mem @ V2A @ ( arr @ A_27c @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) @ V1B ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27c ) @ V2A ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) @ V2A ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EPOW__TWO__set__exp,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ty_2Enum_2Enum ) ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0A ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0A ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eset__exp__count,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ ty_2Enum_2Enum @ A_27a ) @ V0A ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V2l: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) @ V2l )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3e: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2l ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ V0A ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eset__exp__card__cong,axiom,
! [A_27a1: del,A_27a2: del,A_27b1: del,A_27b2: del,V0a1: $i] :
( ( mem @ V0a1 @ ( arr @ A_27a1 @ bool ) )
=> ! [V1a2: $i] :
( ( mem @ V1a2 @ ( arr @ A_27a2 @ bool ) )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ ( arr @ A_27b1 @ bool ) )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ ( arr @ A_27b2 @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a1 @ A_27a2 ) @ V0a1 ) @ V1a2 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b1 @ A_27b2 ) @ V2b1 ) @ V3b2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27b1 @ ( ty_2Eoption_2Eoption @ A_27a1 ) ) @ ( arr @ A_27b2 @ ( ty_2Eoption_2Eoption @ A_27a2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27b1 @ A_27a1 ) @ V0a1 ) @ V2b1 ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27b2 @ A_27a2 ) @ V1a2 ) @ V3b2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eset__exp__cardle__cong,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ bool ) )
=> ! [V1d: $i] :
( ( mem @ V1d @ ( arr @ A_27b @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27c @ bool ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ A_27d @ bool ) )
=> ( ( ( V0b
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ( V1d
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2a ) @ V0b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27d @ A_27b ) @ V3c ) @ V1d ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( arr @ A_27d @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27d @ A_27c ) @ V2a ) @ V3c ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27a ) @ V0b ) @ V1d ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eexp__INSERT__cardeq,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2A: $i] :
( ( mem @ V2A @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ty_2Epair_2Eprod @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ V2A ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) @ V2A ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27b ) @ V2A ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eexp__count__cardeq,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0A ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ ty_2Enum_2Enum @ A_27a ) @ V0A ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ V0A ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__Unum,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0A ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0A ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardleq__SURJ,axiom,
! [A_27a: del,A_27b: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0A ) @ V1B ) )
<=> ( ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27b @ A_27a ) @ V2f ) @ V1B ) @ V0A ) ) )
| ( V0A
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__cardleq__INSERT,axiom,
! [A_27a: del,A_27b: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0A ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V1x ) @ V2s ) ) @ V0A ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V2s ) @ V0A ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Elist__def,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Ecardinal_2Elist @ A_27a ) @ V0A )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V1l: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) @ V1l )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2e: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ V0A ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Elist__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Ecardinal_2Elist @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Elist__SING,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Elist_2Elist @ A_27a ) @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Ecardinal_2Elist @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EUNIV__list,axiom,
! [A_27a: del] :
( ( c_2Epred__set_2EUNIV @ ( ty_2Elist_2Elist @ A_27a ) )
= ( ap @ ( c_2Ecardinal_2Elist @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(conj_thm_2Ecardinal_2Elist__BIGUNION__EXP,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ ( ap @ ( c_2Ecardinal_2Elist @ A_27a ) @ V0A ) )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ ty_2Enum_2Enum @ A_27a ) @ V0A ) @ ( ap @ c_2Epred__set_2Ecount @ V1n ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__A__list__BIJ__A,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0A ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) @ ( ap @ ( c_2Ecardinal_2Elist @ A_27a ) @ V0A ) ) @ V0A ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Efinite__subsets__bijection,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0A ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ ( arr @ A_27a @ bool ) ) @ V0A )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1s ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V0A ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ_27,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0s ) ) )
=> ( ( ( ap @ V1f @ V2x )
= ( ap @ V1f @ V3y ) )
<=> ( V2x = V3y ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ V0s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecountable__decomposition,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ? [V1A: $i] :
( ( mem @ V1A @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1A )
= V0s )
& ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2a ) @ V1A ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2a ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V2a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Edisjoint__countable__decomposition,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ? [V1A: $i] :
( ( mem @ V1A @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
& ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1A )
= V0s )
& ! [V2a: $i] :
( ( mem @ V2a @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2a ) @ V1A ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2a ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V2a ) ) ) ) )
& ! [V3a1: $i] :
( ( mem @ V3a1 @ ( arr @ A_27a @ bool ) )
=> ! [V4a2: $i] :
( ( mem @ V4a2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3a1 ) @ V1A ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4a2 ) @ V1A ) )
& ( V3a1 != V4a2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V3a1 ) @ V4a2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecount__cardle,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ A_27a ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1A ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1A ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1A ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECANTOR,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( arr @ A_27a @ bool ) @ A_27a ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0A ) ) @ V0A ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardlt__cardle,axiom,
! [A_27a: del,A_27b: del,V0B: $i] :
( ( mem @ V0B @ ( arr @ A_27b @ bool ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V0B ) @ V1A ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V1A ) @ V0B ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eset__exp__product,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27c @ bool ) )
=> ! [V1B1: $i] :
( ( mem @ V1B1 @ ( arr @ A_27b @ bool ) )
=> ! [V2B2: $i] :
( ( mem @ V2B2 @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27b @ A_27c ) @ V0A ) @ V1B1 ) ) @ V2B2 ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27c ) @ V0A ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a ) @ V1B1 ) @ V2B2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNT__EQ__EMPTY,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
<=> ( V0n = fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Ecardinal_2EPOW__EQ__X__EXP__X,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0A ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0A ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2Eset__exp @ A_27a @ A_27a ) @ V0A ) @ V0A ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Ebijns__def,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Ecardinal_2Ebijns @ A_27a ) @ V0A )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ^ [V1f: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ bool ) @ V1f )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( c_2Eoption_2ETHE @ A_27a ) ) @ V1f ) ) @ V0A ) @ V0A ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V0A ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3b: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V1f @ V2a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3b ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ecardeq__bijns__cong,axiom,
! [A_27a: del,A_27b: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0A ) @ V1B ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) @ ( ap @ ( c_2Ecardinal_2Ebijns @ A_27a ) @ V0A ) ) @ ( ap @ ( c_2Ecardinal_2Ebijns @ A_27b ) @ V1B ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ebijections__cardeq,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Ecardinal_2Ebijns @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELEFT__IMP__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) )
=> ( p @ V1Q ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELEFT__IMP__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V0P @ V2x ) ) )
=> ( p @ V1Q ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ERIGHT__IMP__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ V0P )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( ( p @ V0P )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ERIGHT__IMP__FORALL__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ V0P )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ V0P )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__FINITE__BIGUNIONS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V0s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECONJ__EQ__IMP,axiom,
! [V0r: $i] :
( ( mem @ V0r @ bool )
=> ! [V1p: $i] :
( ( mem @ V1p @ bool )
=> ! [V2q: $i] :
( ( mem @ V2q @ bool )
=> ( ( ( ( p @ V1p )
& ( p @ V2q ) )
=> ( p @ V0r ) )
<=> ( ( p @ V1p )
=> ( ( p @ V2q )
=> ( p @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EIMP__CONJ__ALT,axiom,
! [V0r: $i] :
( ( mem @ V0r @ bool )
=> ! [V1p: $i] :
( ( mem @ V1p @ bool )
=> ! [V2q: $i] :
( ( mem @ V2q @ bool )
=> ( ( ( ( p @ V1p )
& ( p @ V2q ) )
=> ( p @ V0r ) )
<=> ( ( p @ V2q )
=> ( ( p @ V1p )
=> ( p @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELT__SUC__LE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__ON__RIGHT__INVERSE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V2t ) )
=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( ( ap @ V1f @ V4x )
= V3y ) ) ) )
<=> ? [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27b @ A_27a ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6y ) @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V5g @ V6y ) ) @ V0s ) )
& ( ( ap @ V1f @ ( ap @ V5g @ V6y ) )
= V6y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__RIGHT__INVERSE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ( ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( ( ap @ V0f @ V2x )
= V1y ) ) )
<=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27b ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ap @ V0f @ ( ap @ V3g @ V4y ) )
= V4y ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ__GENERAL,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27b @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) ) )
=> ( V3x = V4y ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1A ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V1A ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27b @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) )
=> ( V2x = V3y ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1A ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V4x ) ) @ V1A ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) ) )
=> ( V2x = V3y ) ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__IMAGE__INJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V2y ) )
=> ( V1x = V2y ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__NONEMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__PRODUCT__DEPENDENT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ V2t @ V3x ) ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27c @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27c @ bool ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( lam @ A_27b
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ bool ) @ ( ap @ ( ap @ V0f @ V4x ) @ V5y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ ( ap @ V2t @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__PRODUCT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__IMAGE__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( ( ap @ V0f @ V2x )
= V1y ) ) )
<=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ V3P @ ( ap @ V0f @ V4x ) ) ) ) ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V5x ) @ ( ap @ V3P @ V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__ON__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( arr @ A_27b @ bool ) )
=> ( ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V3t ) @ V2v ) )
=> ? [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V4s ) @ V1u ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V4s )
= V3t ) ) ) )
<=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V2v ) )
=> ? [V6x: $i] :
( ( mem @ V6x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V1u ) )
& ( ( ap @ V0f @ V6x )
= V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s )
= V1t ) ) )
<=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( ( ap @ V0f @ V4x )
= V3y ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__INJ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V0s ) ) @ V1t ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V0s ) )
& ( ( ap @ V2f @ V3x )
= ( ap @ V2f @ V4y ) ) )
=> ( V3x = V4y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__IMAGE__INJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) ) )
=> ( V2x = V3y ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__IMAGE__LE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE__GEN,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V0s ) ) @ V1t ) ) )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) )
=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( ( ap @ V2f @ V4x )
= V3y ) ) ) )
<=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6y ) @ V0s ) )
& ( ( ap @ V2f @ V5x )
= ( ap @ V2f @ V6y ) ) )
=> ( V5x = V6y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V0s ) ) @ V0s ) ) )
=> ( ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V0s ) )
=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
& ( ( ap @ V1f @ V3x )
= V2y ) ) ) )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V0s ) )
& ( ( ap @ V1f @ V4x )
= ( ap @ V1f @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__BIJECTION,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1t ) ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V1t ) )
=> ? [V5x: $i] :
( ( mem @ V5x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) )
& ( ( ap @ V2f @ V5x )
= V4y ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7y ) @ V0s ) )
& ( ( ap @ V2f @ V6x )
= ( ap @ V2f @ V7y ) ) )
=> ( V6x = V7y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__BIJECTIONS,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V4x ) ) @ V1t ) )
& ( ( ap @ V3g @ ( ap @ V2f @ V4x ) )
= V4x ) ) ) )
& ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V3g @ V5y ) ) @ V0s ) )
& ( ( ap @ V2f @ ( ap @ V3g @ V5y ) )
= V5y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ESING__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINJECTIVE__ON__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1u: $i] :
( ( mem @ V1u @ ( arr @ A_27a @ bool ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s ) @ V1u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ V1u ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3t ) ) )
=> ( V2s = V3t ) ) ) )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V1u ) )
& ( ( ap @ V0f @ V4x )
= ( ap @ V0f @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINJECTIVE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) )
=> ( V1s = V2t ) ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) )
=> ( V3x = V4y ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__FINITE__BIGUNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V0s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Enum__FINITE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
<=> ? [V1a: tp__ty_2Enum_2Enum] :
! [V2x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Enum__FINITE__AVOID,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ? [V1a: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Enum__INFINITE,axiom,
~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ).
thf(ax_thm_2Ecardinal_2EHAS__SIZE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= V1n ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__CARD,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= V1n ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__0,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__SUC,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__HAS__SIZE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__CLAUSES,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V1s ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V1s ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
<=> ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V3t ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ V3t ) )
& ( V1s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2a ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__SUBSET__EQ,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1b ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0a ) @ V1b ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0a ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1b ) ) ) )
=> ( V0a = V1b ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__INDEX,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ V0s ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5m: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V5m ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2f @ V5m ) ) @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__BIGUNION__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ V1t @ V4x ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ V1t @ V4x ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ c_2Earithmetic_2E_3C_3D
@ ( ap @ ( c_2Epred__set_2ECARD @ A_27b )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27b @ bool ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ bool ) @ ( ap @ V1t @ V5x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__BOOL,axiom,
p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ bool ) @ ( c_2Epred__set_2EUNIV @ bool ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__BOOL,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ bool ) @ ( c_2Epred__set_2EUNIV @ bool ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__BOOL,axiom,
p @ ( ap @ ( c_2Epred__set_2EFINITE @ bool ) @ ( c_2Epred__set_2EUNIV @ bool ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__CART__UNIV,axiom,
! [A_27a: del,V0m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__CART__UNIV,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__CART__UNIV,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LT,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1m ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V1m ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__NUMSEG__LT,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1m ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V1m ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LE,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1m ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V1m ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) )
@ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__NUMSEG__LE,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1m ) @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V1m ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__DIFF__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELE__CASES,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
| ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELT__CASES,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
| ( V0m = V1n ) ) ).
thf(conj_thm_2Ecardinal_2ELT,axiom,
( ! [V0m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
<=> $false )
& ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
<=> ( ( V1m = V2n )
| ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELT__LE,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
& ( V0m != V1n ) ) ) ).
thf(conj_thm_2Ecardinal_2EGE,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3E_3D @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELE__SUC__LT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINJECTIVE__ON__LEFT__INVERSE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) ) )
=> ( V2x = V3y ) ) ) )
<=> ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27b @ A_27a ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1s ) )
=> ( ( ap @ V4g @ ( ap @ V0f @ V5x ) )
= V5x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINJECTIVE__LEFT__INVERSE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V2y ) )
=> ( V1x = V2y ) ) ) )
<=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ap @ V3g @ ( ap @ V0f @ V4x ) )
= V4x ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINTER__ACI,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V2q )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2q ) @ V1p ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V2q ) ) @ V0r )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V2q ) ) @ V0r ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V2q ) ) @ V0r )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V2q ) @ V1p ) ) @ V0r ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V1p )
= V1p )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V1p ) ) @ V2q )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1p ) @ V2q ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECONJ__ACI,axiom,
! [V0r: $i] :
( ( mem @ V0r @ bool )
=> ! [V1p: $i] :
( ( mem @ V1p @ bool )
=> ! [V2q: $i] :
( ( mem @ V2q @ bool )
=> ( ( ( ( p @ V1p )
& ( p @ V2q ) )
<=> ( ( p @ V2q )
& ( p @ V1p ) ) )
& ( ( ( p @ V1p )
& ( p @ V2q )
& ( p @ V0r ) )
<=> ( ( p @ V1p )
& ( p @ V2q )
& ( p @ V0r ) ) )
& ( ( ( p @ V1p )
& ( p @ V2q )
& ( p @ V0r ) )
<=> ( ( p @ V2q )
& ( p @ V1p )
& ( p @ V0r ) ) )
& ( ( ( p @ V1p )
& ( p @ V1p ) )
<=> ( p @ V1p ) )
& ( ( ( p @ V1p )
& ( p @ V1p )
& ( p @ V2q ) )
<=> ( ( p @ V1p )
& ( p @ V2q ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EUNION__ACI,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ! [V2q: $i] :
( ( mem @ V2q @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V2q )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2q ) @ V1p ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V2q ) ) @ V0r )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V2q ) ) @ V0r ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V2q ) ) @ V0r )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V2q ) @ V1p ) ) @ V0r ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V1p )
= V1p )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V1p ) ) @ V2q )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1p ) @ V2q ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELT__NZ,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
<=> ( V0n != fo__c_2Enum_2E0 ) ) ).
thf(conj_thm_2Ecardinal_2ELE__1,axiom,
( ! [V0n: tp__ty_2Enum_2Enum] :
( ( V0n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum] :
( ( V1n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( V2n != fo__c_2Enum_2E0 ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) )
=> ( V5n != fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Ecardinal_2EOR__EXISTS__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0P @ V2x ) ) )
| ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V1Q @ V3x ) ) ) )
<=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( ( p @ ( ap @ V0P @ V4x ) )
| ( p @ ( ap @ V1Q @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EBIJECTIVE__INJECTIVE__SURJECTIVE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V2t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V4y ) ) ) ) ) ) ) )
<=> ( ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V6x ) ) @ V2t ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8y ) @ V1s ) )
& ( ( ap @ V0f @ V7x )
= ( ap @ V0f @ V8y ) ) )
=> ( V7x = V8y ) ) ) )
& ! [V9y: $i] :
( ( mem @ V9y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V9y ) @ V2t ) )
=> ? [V10x: $i] :
( ( mem @ V10x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V10x ) @ V1s ) )
& ( ( ap @ V0f @ V10x )
= V9y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EBIJECTIVE__INVERSES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V2t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V4y ) ) ) ) ) ) ) )
<=> ( ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V6x ) ) @ V2t ) ) ) )
& ? [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27b @ A_27a ) )
& ! [V8y: $i] :
( ( mem @ V8y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V8y ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V7g @ V8y ) ) @ V1s ) ) ) )
& ! [V9y: $i] :
( ( mem @ V9y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V9y ) @ V2t ) )
=> ( ( ap @ V0f @ ( ap @ V7g @ V9y ) )
= V9y ) ) )
& ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V10x ) @ V1s ) )
=> ( ( ap @ V7g @ ( ap @ V0f @ V10x ) )
= V10x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ele__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1t ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V0s ) )
& ( ( ap @ V2f @ V4x )
= ( ap @ V2f @ V5y ) ) )
=> ( V4x = V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Elt__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Eeq__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1t ) ) ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V1t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V2f @ V5x ) ) @ V4y ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Ecardgeq__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Ege__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Ecardgt__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardgt @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Egt__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardgt @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ELE__C,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ? [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27a ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
=> ? [V4y: $i] :
( ( mem @ V4y @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V1t ) )
& ( ( ap @ V2g @ V4y )
= V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EGE__C,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardgeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) )
=> ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0s ) )
& ( V3y
= ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardgeq @ ty_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0t ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EEQ__C,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ? [V2R: $i] :
( ( mem @ V2R @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ V2R @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V1t ) ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V0s ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27b )
@ ( lam @ A_27b
@ ^ [V6y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6y ) @ V1t ) ) @ ( ap @ V2R @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V5x ) @ V6y ) ) ) ) ) ) ) )
& ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7y ) @ V1t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V0s ) ) @ ( ap @ V2R @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V8x ) @ V7y ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LET__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V2u ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2u ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LTE__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2u ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V2u ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2u ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__SYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__IMP__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__IMP__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__RELATIONAL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ! [V4y_27: $i] :
( ( mem @ V4y_27 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V4y_27 ) ) )
=> ( V3y = V4y_27 ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V5y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V5y )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V1s ) ) @ ( ap @ ( ap @ V0R @ V6x ) @ V5y ) ) ) ) ) ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__ANTISYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__TOTAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LET__TOTAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
| ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LTE__TOTAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__TOTAL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
| ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
| ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__NOT__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__NOT__LT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__LT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V0s ) @ V2t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2t ) @ V0s ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27d @ A_27b ) @ V3t_27 ) @ V1s_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27c ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27c ) @ V0s ) @ V2t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EINFINITE__CARD__LE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__CARD__LT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ty_2Enum_2Enum @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__UNIV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__EQ__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ? [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27b @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V2u ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__INFINITE__CONG,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__FINITE__CONG,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__INFINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__FINITE__INFINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__CARD__IMP,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__CARD__IMP,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__CARD,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__CARD,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__CARD,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__HAS__SIZE__CONG,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2EHAS__SIZE @ A_27b ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__IMAGE__GEN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V2t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V2t ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) ) )
=> ( V2x = V3y ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2Ecardinal_2Eadd__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t )
= ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b )
@ ( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3y ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2Emul__c,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__ADD,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ A_27a @ A_27c ) @ ( ty_2Esum_2Esum @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27c ) @ V0s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__MUL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c ) @ V0s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Esum_2Esum @ A_27a @ A_27c ) @ ( ty_2Esum_2Esum @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27c ) @ V0s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27c @ A_27d ) @ V2t ) @ V3t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c ) @ V0s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EIN__CARD__ADD,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V3y ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EIN__CARD__MUL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__SQUARE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__SQUARE__NUM,axiom,
p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Ecardinal_2EUNION__LE__ADD__C,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__C,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__SYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( ty_2Esum_2Esum @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Esum_2Esum @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27c ) ) @ ( ty_2Esum_2Esum @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27c ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__SYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LDISTRIB,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27c ) ) @ ( ty_2Esum_2Esum @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27c ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c ) @ V0s ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__RDISTRIB,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) @ ( ty_2Esum_2Esum @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V2u ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27c ) @ V0s ) @ V2u ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__ADDR,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__ADDL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ V1t ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__LE__MUL__INFINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__DISJOINT__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__SQUARE__INFINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V0s ) @ V0s ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__FINITE__EQ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__ABSORB__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL2__ABSORB__LE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__ABSORB__LE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD2__ABSORB__LE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27c ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27c ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__ABSORB,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD__ABSORB,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__ADD2__ABSORB__LT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ V2u ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2u ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V2u ) @ V1t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ V2u ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LT__ADD,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1s_27: $i] :
( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27c @ bool ) )
=> ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27d @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1s_27 ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27d @ A_27c ) @ V3t_27 ) @ V2t ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Esum_2Esum @ A_27b @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27b @ A_27d ) @ V1s_27 ) @ V3t_27 ) ) @ ( ap @ ( ap @ ( c_2Ecardinal_2E_2B__c @ A_27a @ A_27c ) @ V0s ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__LT__LEMMA,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V2u ) @ V1t ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ V2u ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2u ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__MUL__LT__INFINITE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27c @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27a ) @ V2u ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ A_27b ) @ V2u ) @ V1t ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27c ) @ V2u ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27c @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2u ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECANTOR__THM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ( p
@ ( ap
@ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( arr @ A_27a @ bool ) @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1t ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) ) ) )
@ V0s ) ) ) ).
thf(conj_thm_2Ecardinal_2ECANTOR__THM__UNIV,axiom,
! [A_27a: del] :
~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( arr @ A_27a @ bool ) @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(conj_thm_2Ecardinal_2ENUM__COUNTABLE,axiom,
p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__ALT__cardleq,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ ty_2Enum_2Enum ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__CASES,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ ty_2Enum_2Enum ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__LE__COUNTABLE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27a ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__EQ__COUNTABLE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECARD__COUNTABLE__CONG,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardeq @ A_27a @ A_27a ) @ V0s ) @ V1t ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__RESTRICT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ V1P @ V2x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFINITE__IMP__COUNTABLE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__IMAGE__INJ__GENERAL,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27b @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) ) )
=> ( V3x = V4y ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V1A ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V5x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ V1A ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__IMAGE__INJ__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) ) )
=> ( V2x = V3y ) ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__IMAGE__INJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1A: $i] :
( ( mem @ V1A @ ( arr @ A_27b @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) )
=> ( V2x = V3y ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V1A ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V4x ) ) @ V1A ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__EMPTY,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__INTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
| ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__UNION__IMP,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__DIFF__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1t ) @ V0s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__CROSS,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__AS__IMAGE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__AS__IMAGE__SUBSET__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__AS__IMAGE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EFORALL__COUNTABLE__AS__IMAGE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V1d: $i] :
( ( mem @ V1d @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1d ) )
=> ( p @ ( ap @ V0P @ V1d ) ) ) )
<=> ( ( p @ ( ap @ V0P @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V2f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__AS__INJECTIVE__IMAGE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) )
& ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2m ) )
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( V2m = V3n ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__BIGUNION,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V0A ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ V0A ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0A ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2EIN__ELIM__PAIR__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1a ) @ V2b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) @ ( ap @ ( ap @ V0P @ V3x ) @ V4y ) ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ V0P @ V1a ) @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Ecardinal_2ECOUNTABLE__PRODUCT__DEPENDENT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( ap @ V2t @ V3x ) ) ) ) ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2Ecountable @ A_27c )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27c @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27c @ bool ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( lam @ A_27b
@ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ bool ) @ ( ap @ ( ap @ V0f @ V4x ) @ V5y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5y ) @ ( ap @ V2t @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------