TPTP Problem File: ITP010+5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP010+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ecardinal_2ECARD__NOT__LE.p, 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 : thm_2Ecardinal_2ECARD__NOT__LE.p [Gau20]
% : HL404501+5.p [TPAP]
% Status : Theorem
% Rating : 0.47 v8.2.0, 0.44 v7.5.0
% Syntax : Number of formulae : 8598 ( 455 unt; 0 def)
% Number of atoms : 62814 (10133 equ)
% Maximal formula atoms : 5765 ( 7 avg)
% Number of connectives : 55051 ( 835 ~; 457 |;19057 &)
% (3601 <=>;31101 =>; 0 <=; 0 <~>)
% Maximal formula depth : 363 ( 9 avg)
% Maximal term depth : 28 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 2839 (2839 usr; 283 con; 0-11 aty)
% Number of variables : 44094 (31308 !;12786 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
include('Axioms/ITP001/ITP002+5.ax').
include('Axioms/ITP001/ITP003+5.ax').
include('Axioms/ITP001/ITP004+5.ax').
include('Axioms/ITP001/ITP007+5.ax').
include('Axioms/ITP001/ITP006+5.ax').
include('Axioms/ITP001/ITP005+5.ax').
include('Axioms/ITP001/ITP008+5.ax').
include('Axioms/ITP001/ITP009+5.ax').
include('Axioms/ITP001/ITP010+5.ax').
include('Axioms/ITP001/ITP012+5.ax').
include('Axioms/ITP001/ITP011+5.ax').
include('Axioms/ITP001/ITP013+5.ax').
include('Axioms/ITP001/ITP014+5.ax').
include('Axioms/ITP001/ITP015+5.ax').
include('Axioms/ITP001/ITP017+5.ax').
include('Axioms/ITP001/ITP016+5.ax').
include('Axioms/ITP001/ITP019+5.ax').
include('Axioms/ITP001/ITP018+5.ax').
include('Axioms/ITP001/ITP021+5.ax').
include('Axioms/ITP001/ITP022+5.ax').
include('Axioms/ITP001/ITP020+5.ax').
include('Axioms/ITP001/ITP024+5.ax').
include('Axioms/ITP001/ITP023+5.ax').
include('Axioms/ITP001/ITP025+5.ax').
include('Axioms/ITP001/ITP026+5.ax').
include('Axioms/ITP001/ITP027+5.ax').
include('Axioms/ITP001/ITP028+5.ax').
include('Axioms/ITP001/ITP031+5.ax').
include('Axioms/ITP001/ITP029+5.ax').
include('Axioms/ITP001/ITP033+5.ax').
include('Axioms/ITP001/ITP030+5.ax').
include('Axioms/ITP001/ITP032+5.ax').
include('Axioms/ITP001/ITP038+5.ax').
include('Axioms/ITP001/ITP035+5.ax').
include('Axioms/ITP001/ITP034+5.ax').
include('Axioms/ITP001/ITP036+5.ax').
include('Axioms/ITP001/ITP037+5.ax').
include('Axioms/ITP001/ITP039+5.ax').
include('Axioms/ITP001/ITP041+5.ax').
include('Axioms/ITP001/ITP042+5.ax').
include('Axioms/ITP001/ITP040+5.ax').
include('Axioms/ITP001/ITP044+5.ax').
include('Axioms/ITP001/ITP051+5.ax').
include('Axioms/ITP001/ITP045+5.ax').
include('Axioms/ITP001/ITP056+5.ax').
include('Axioms/ITP001/ITP046+5.ax').
include('Axioms/ITP001/ITP043+5.ax').
include('Axioms/ITP001/ITP052+5.ax').
include('Axioms/ITP001/ITP057+5.ax').
include('Axioms/ITP001/ITP048+5.ax').
include('Axioms/ITP001/ITP047+5.ax').
include('Axioms/ITP001/ITP055+5.ax').
include('Axioms/ITP001/ITP053+5.ax').
include('Axioms/ITP001/ITP054+5.ax').
include('Axioms/ITP001/ITP058+5.ax').
include('Axioms/ITP001/ITP049+5.ax').
include('Axioms/ITP001/ITP050+5.ax').
include('Axioms/ITP001/ITP061+5.ax').
include('Axioms/ITP001/ITP069+5.ax').
include('Axioms/ITP001/ITP062+5.ax').
include('Axioms/ITP001/ITP068+5.ax').
include('Axioms/ITP001/ITP078+5.ax').
include('Axioms/ITP001/ITP064+5.ax').
include('Axioms/ITP001/ITP060+5.ax').
include('Axioms/ITP001/ITP067+5.ax').
include('Axioms/ITP001/ITP075+5.ax').
include('Axioms/ITP001/ITP074+5.ax').
%------------------------------------------------------------------------------
fof(mem_c_2Ecardinal_2E_2B__c,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Ecardinal_2EHAS__SIZE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecardinal_2EHAS__SIZE(A_27a),arr(arr(A_27a,bool),arr(ty_2Enum_2Enum,bool))) ) ).
fof(mem_c_2Ecardinal_2Ebijns,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecardinal_2Ebijns(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,ty_2Eoption_2Eoption(A_27a)),bool))) ) ).
fof(mem_c_2Ecardinal_2Ecardeq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecardinal_2Ecardeq(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) ).
fof(mem_c_2Ecardinal_2Ecardgeq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecardinal_2Ecardgeq(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) ).
fof(mem_c_2Ecardinal_2Ecardgt,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecardinal_2Ecardgt(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) ).
fof(mem_c_2Ecardinal_2Ecardleq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecardinal_2Ecardleq(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool))) ) ) ).
fof(mem_c_2Ecardinal_2Elist,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecardinal_2Elist(A_27a),arr(arr(A_27a,bool),arr(ty_2Elist_2Elist(A_27a),bool))) ) ).
fof(mem_c_2Ecardinal_2Eset__exp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(ax_thm_2Ecardinal_2Ecardeq__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s1),V1s2)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardeq__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardeq__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardeq__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Ecardinal_2Ecardleq__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s1),V1s2))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s1),V1s2)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__ANTISYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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)) )
=> ap(c_2Epred__set_2ECARD(A_27a),V0s1) = ap(c_2Epred__set_2ECARD(A_27b),V1s2) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0x] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardeq__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__INSERT__RWT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EEMPTY__CARDLEQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t] :
( mem(V0t,arr(A_27b,bool))
=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),c_2Epred__set_2EEMPTY(A_27a)),V0t)) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__CLE__INFINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__CROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ! [V2t1] :
( mem(V2t1,arr(A_27c,bool))
=> ! [V3t2] :
( 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))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__CROSS__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__SUBSET__CARDLEQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__CARDLEQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ! [V2t1] :
( mem(V2t1,arr(A_27c,bool))
=> ! [V3t2] :
( 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)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDLEQ__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__UNIV__INF,axiom,
! [A_27a] :
( ne(A_27a)
=> ~ 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)))) ) ).
fof(conj_thm_2Ecardinal_2EIMAGE__cardleq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDLEQ__CROSS__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0x1] :
( mem(V0x1,arr(A_27a,bool))
=> ! [V1x2] :
( mem(V1x2,arr(A_27b,bool))
=> ! [V2y1] :
( mem(V2y1,arr(A_27c,bool))
=> ! [V3y2] :
( 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))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESUBSET__CARDLEQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EIMAGE__cardleq__rwt,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecountable__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecountable__cardeq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__dichotomy,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__lteq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardlt__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ~ ~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardlt__lenoteq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardlt__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27b,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27c,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardlt__leq__trans,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0r] :
( mem(V0r,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__lt__trans,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0r] :
( mem(V0r,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__empty,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( 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) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eset__binomial2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( 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)) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESET__SQUARED__CARDEQ__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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)) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESET__SUM__CARDEQ__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( 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] :
( mem(V1A,arr(A_27b,bool))
=> ! [V2B] :
( 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)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0k] :
( mem(V0k,arr(A_27a,bool))
=> ! [V1s1] :
( 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] :
( 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)) ) ) ) ) ) ).
fof(lameq_f1848,axiom,
! [A_27a,A_27b,V3b] :
( mem(V3b,A_27a)
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eoption_2Eoption(A_27b)))
=> ! [V0A] :
( mem(V0A,arr(A_27b,bool))
=> ! [V4a] : ap(f1848(A_27a,A_27b,V3b,V2f,V0A),V4a) = 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))) ) ) ) ).
fof(lameq_f1849,axiom,
! [A_27b,A_27a,V2f] :
( mem(V2f,arr(A_27a,ty_2Eoption_2Eoption(A_27b)))
=> ! [V0A] :
( mem(V0A,arr(A_27b,bool))
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V3b] : ap(f1849(A_27b,A_27a,V2f,V0A,V1B),V3b) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(A_27a),V3b),V1B)),ap(c_2Ebool_2E_3F(A_27b),f1848(A_27a,A_27b,V3b,V2f,V0A))) ) ) ) ).
fof(lameq_f1850,axiom,
! [A_27b,A_27a,V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Eoption_2Eoption(A_27b)))
=> ! [V5b] : ap(f1850(A_27b,A_27a,V1B,V2f),V5b) = 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))) ) ) ).
fof(lameq_f1851,axiom,
! [A_27b,A_27a,V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V0A] :
( mem(V0A,arr(A_27b,bool))
=> ! [V2f] : ap(f1851(A_27b,A_27a,V1B,V0A),V2f) = 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),f1849(A_27b,A_27a,V2f,V0A,V1B))),ap(c_2Ebool_2E_21(A_27a),f1850(A_27b,A_27a,V1B,V2f)))) ) ) ).
fof(ax_thm_2Ecardinal_2Eset__exp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0A] :
( mem(V0A,arr(A_27b,bool))
=> ! [V1B] :
( 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))),f1851(A_27b,A_27a,V1B,V0A)) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EBIJ__functions__agree,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V2s),V3t))
& ! [V4x] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__CARDEQ__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s1))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s2))
& ap(c_2Epred__set_2ECARD(A_27a),V0s1) = ap(c_2Epred__set_2ECARD(A_27b),V1s2) )
=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s1),V1s2)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDEQ__CARD__EQN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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))
<=> ap(c_2Epred__set_2ECARD(A_27a),V0s1) = ap(c_2Epred__set_2ECARD(A_27b),V1s2) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARDLEQ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EEMPTY__set__exp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0A] :
( mem(V0A,arr(A_27b,bool))
=> ! [V1B] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EEMPTY__set__exp__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0A] :
( 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,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(lameq_f1852,axiom,
! [A_27b,A_27a,V0x] :
( mem(V0x,A_27b)
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V3b] : ap(f1852(A_27b,A_27a,V0x,V1B),V3b) = 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)) ) ) ).
fof(lameq_f1853,axiom,
! [A_27c,A_27b,V4a] :
( mem(V4a,A_27c)
=> ! [V0x] :
( mem(V0x,A_27b)
=> ! [V5b] : ap(f1853(A_27c,A_27b,V4a,V0x),V5b) = 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)) ) ) ).
fof(lameq_f1854,axiom,
! [A_27b,A_27c,V0x] :
( mem(V0x,A_27b)
=> ! [V2A] :
( mem(V2A,arr(A_27c,bool))
=> ! [V4a] : ap(f1854(A_27b,A_27c,V0x,V2A),V4a) = ap(ap(c_2Epair_2E_2C(arr(A_27b,ty_2Eoption_2Eoption(A_27c)),bool),f1853(A_27c,A_27b,V4a,V0x)),ap(ap(c_2Ebool_2EIN(A_27c),V4a),V2A)) ) ) ).
fof(conj_thm_2Ecardinal_2ESING__set__exp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0x] :
( mem(V0x,A_27b)
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V2A] :
( 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))),f1852(A_27b,A_27a,V0x,V1B)),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),f1854(A_27b,A_27c,V0x,V2A)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESING__set__exp__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0x] :
( mem(V0x,A_27b)
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V2A] :
( 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,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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EPOW__TWO__set__exp,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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,c_2Earithmetic_2EZERO)))),V0A))) ) ) ).
fof(lameq_f1855,axiom,
! [A_27a,V2l] :
( mem(V2l,ty_2Elist_2Elist(A_27a))
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V3e] : ap(f1855(A_27a,V2l,V0A),V3e) = 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)) ) ) ).
fof(lameq_f1856,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2l] : ap(f1856(A_27a,V0A,V1n),V2l) = 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)),V1n)),ap(c_2Ebool_2E_21(A_27a),f1855(A_27a,V2l,V0A)))) ) ) ).
fof(conj_thm_2Ecardinal_2Eset__exp__count,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,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,V1n))),ap(c_2Epred__set_2EGSPEC(ty_2Elist_2Elist(A_27a),ty_2Elist_2Elist(A_27a)),f1856(A_27a,V0A,V1n)))) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eset__exp__card__cong,axiom,
! [A_27a1] :
( ne(A_27a1)
=> ! [A_27a2] :
( ne(A_27a2)
=> ! [A_27b1] :
( ne(A_27b1)
=> ! [A_27b2] :
( ne(A_27b2)
=> ! [V0a1] :
( mem(V0a1,arr(A_27a1,bool))
=> ! [V1a2] :
( mem(V1a2,arr(A_27a2,bool))
=> ! [V2b1] :
( mem(V2b1,arr(A_27b1,bool))
=> ! [V3b2] :
( 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))) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eset__exp__cardle__cong,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0b] :
( mem(V0b,arr(A_27a,bool))
=> ! [V1d] :
( mem(V1d,arr(A_27b,bool))
=> ! [V2a] :
( mem(V2a,arr(A_27c,bool))
=> ! [V3c] :
( 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))) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eexp__INSERT__cardeq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2A] :
( 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)))) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eexp__count__cardeq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0A))
& p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),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,V1n))),V0A)) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__Unum,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardleq__SURJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( mem(V1B,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0A),V1B))
<=> ( ? [V2f] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__cardleq__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27b)
=> ! [V2s] :
( 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)) ) ) ) ) ) ) ) ).
fof(lameq_f1857,axiom,
! [A_27a,V1l] :
( mem(V1l,ty_2Elist_2Elist(A_27a))
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V2e] : ap(f1857(A_27a,V1l,V0A),V2e) = 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)) ) ) ).
fof(lameq_f1858,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1l] : ap(f1858(A_27a,V0A),V1l) = ap(ap(c_2Epair_2E_2C(ty_2Elist_2Elist(A_27a),bool),V1l),ap(c_2Ebool_2E_21(A_27a),f1857(A_27a,V1l,V0A))) ) ).
fof(ax_thm_2Ecardinal_2Elist__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)),f1858(A_27a,V0A)) ) ) ).
fof(conj_thm_2Ecardinal_2Elist__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> 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))) ) ).
fof(conj_thm_2Ecardinal_2Elist__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( 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))) ) ) ).
fof(conj_thm_2Ecardinal_2EUNIV__list,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epred__set_2EUNIV(ty_2Elist_2Elist(A_27a)) = ap(c_2Ecardinal_2Elist(A_27a),c_2Epred__set_2EUNIV(A_27a)) ) ).
fof(lameq_f1859,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1n] : ap(f1859(A_27a,V0A),V1n) = ap(ap(c_2Ecardinal_2Eset__exp(ty_2Enum_2Enum,A_27a),V0A),ap(c_2Epred__set_2Ecount,V1n)) ) ).
fof(conj_thm_2Ecardinal_2Elist__BIGUNION__EXP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)),f1859(A_27a,V0A)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__A__list__BIJ__A,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)) ) ) ) ).
fof(lameq_f1860,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1s] : ap(f1860(A_27a,V0A),V1s) = 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))) ) ).
fof(conj_thm_2Ecardinal_2Efinite__subsets__bijection,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)),f1860(A_27a,V0A)))) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ_27,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecountable__decomposition,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ? [V1A] :
( mem(V1A,arr(arr(A_27a,bool),bool))
& ap(c_2Epred__set_2EBIGUNION(A_27a),V1A) = V0s
& ! [V2a] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Edisjoint__countable__decomposition,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ? [V1A] :
( mem(V1A,arr(arr(A_27a,bool),bool))
& ap(c_2Epred__set_2EBIGUNION(A_27a),V1A) = V0s
& ! [V2a] :
( 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] :
( mem(V3a1,arr(A_27a,bool))
=> ! [V4a2] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ecount__cardle,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1A] :
( mem(V1A,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Enum_2Enum,A_27a),ap(c_2Epred__set_2Ecount,V0n)),V1A))
<=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1A))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),ap(c_2Epred__set_2ECARD(A_27a),V1A))) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECANTOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardlt__cardle,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0B] :
( mem(V0B,arr(A_27b,bool))
=> ! [V1A] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eset__exp__product,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0A] :
( mem(V0A,arr(A_27c,bool))
=> ! [V1B1] :
( mem(V1B1,arr(A_27b,bool))
=> ! [V2B2] :
( 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)))) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECOUNT__EQ__EMPTY,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( ap(c_2Epred__set_2Ecount,V0n) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)
<=> V0n = c_2Enum_2E0 ) ) ).
fof(conj_thm_2Ecardinal_2EPOW__EQ__X__EXP__X,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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))) ) ) ) ).
fof(lameq_f1861,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(A_27a)))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] : ap(f1861(A_27a,V1f,V2a),V3b) = ap(ap(c_2Emin_2E_3D(ty_2Eoption_2Eoption(A_27a)),ap(V1f,V2a)),ap(c_2Eoption_2ESOME(A_27a),V3b)) ) ) ).
fof(lameq_f1862,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Eoption_2Eoption(A_27a)))
=> ! [V2a] : ap(f1862(A_27a,V0A,V1f),V2a) = ap(ap(c_2Emin_2E_3D(bool),ap(ap(c_2Ebool_2EIN(A_27a),V2a),V0A)),ap(c_2Ebool_2E_3F(A_27a),f1861(A_27a,V1f,V2a))) ) ) ).
fof(lameq_f1863,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1f] : ap(f1863(A_27a,V0A),V1f) = 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),f1862(A_27a,V0A,V1f)))) ) ).
fof(ax_thm_2Ecardinal_2Ebijns__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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))),f1863(A_27a,V0A)) ) ) ).
fof(conj_thm_2Ecardinal_2Ecardeq__bijns__cong,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ebijections__cardeq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELEFT__IMP__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
=> p(V1Q) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
=> p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELEFT__IMP__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V0P,V2x)) )
=> p(V1Q) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(ap(V0P,V3x))
=> p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ERIGHT__IMP__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
=> ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V1Q,V2x)) ) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(V0P)
=> p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ERIGHT__IMP__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
=> ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V1Q,V2x)) ) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(V0P)
=> p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__FINITE__BIGUNIONS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECONJ__EQ__IMP,axiom,
! [V0r] :
( mem(V0r,bool)
=> ! [V1p] :
( mem(V1p,bool)
=> ! [V2q] :
( mem(V2q,bool)
=> ( ( ( p(V1p)
& p(V2q) )
=> p(V0r) )
<=> ( p(V1p)
=> ( p(V2q)
=> p(V0r) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EIMP__CONJ__ALT,axiom,
! [V0r] :
( mem(V0r,bool)
=> ! [V1p] :
( mem(V1p,bool)
=> ! [V2q] :
( mem(V2q,bool)
=> ( ( ( p(V1p)
& p(V2q) )
=> p(V0r) )
<=> ( p(V2q)
=> ( p(V1p)
=> p(V0r) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELT__SUC__LE,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),ap(c_2Enum_2ESUC,V1n)))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__ON__RIGHT__INVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ! [V3y] :
( mem(V3y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V2t))
=> ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
& ap(V1f,V4x) = V3y ) ) )
<=> ? [V5g] :
( mem(V5g,arr(A_27b,A_27a))
& ! [V6y] :
( 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 ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__RIGHT__INVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,A_27a))
=> ( ! [V1y] :
( mem(V1y,A_27a)
=> ? [V2x] :
( mem(V2x,A_27b)
& ap(V0f,V2x) = V1y ) )
<=> ? [V3g] :
( mem(V3g,arr(A_27a,A_27b))
& ! [V4y] :
( mem(V4y,A_27a)
=> ap(V0f,ap(V3g,V4y)) = V4y ) ) ) ) ) ) ).
fof(lameq_f1864,axiom,
! [A_27b,A_27a,V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1A] :
( mem(V1A,arr(A_27b,bool))
=> ! [V5x] : ap(f1864(A_27b,A_27a,V2s,V0f,V1A),V5x) = 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))) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ__GENERAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1A] :
( mem(V1A,arr(A_27b,bool))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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),f1864(A_27b,A_27a,V2s,V0f,V1A)))) ) ) ) ) ) ) ).
fof(lameq_f1865,axiom,
! [A_27b,A_27a,V1A] :
( mem(V1A,arr(A_27b,bool))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V4x] : ap(f1865(A_27b,A_27a,V1A,V0f),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V4x)),V1A)) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1A] :
( mem(V1A,arr(A_27b,bool))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( 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),f1865(A_27b,A_27a,V1A,V0f)))) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__IMAGE__INJ__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__IMAGE__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ap(V0f,V1x) = ap(V0f,V2y)
=> V1x = V2y ) ) )
=> ! [V3s] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__NONEMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> V0s != c_2Epred__set_2EEMPTY(A_27a) ) ) ) ).
fof(lameq_f1866,axiom,
! [A_27a,A_27c,A_27b,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] : ap(f1866(A_27a,A_27c,A_27b,V1s,V2t,V0f,V4x),V5y) = 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)))) ) ) ) ) ).
fof(lameq_f1867,axiom,
! [A_27b,A_27c,A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V4x] : ap(f1867(A_27b,A_27c,A_27a,V1s,V2t,V0f),V4x) = f1866(A_27a,A_27c,A_27b,V1s,V2t,V0f,V4x) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__PRODUCT__DEPENDENT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,arr(A_27b,bool)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& ! [V3x] :
( 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)),f1867(A_27b,A_27c,A_27a,V1s,V2t,V0f))))) ) ) ) ) ) ) ) ).
fof(lameq_f1868,axiom,
! [A_27a,A_27b,V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] : ap(f1868(A_27a,A_27b,V1t,V0s,V2x),V3y) = 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))) ) ) ) ).
fof(lameq_f1869,axiom,
! [A_27b,A_27a,V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x] : ap(f1869(A_27b,A_27a,V1t,V0s),V2x) = f1868(A_27a,A_27b,V1t,V0s,V2x) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__PRODUCT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)),f1869(A_27b,A_27a,V1t,V0s))))) ) ) ) ) ) ).
fof(lameq_f1870,axiom,
! [A_27b,A_27a,V3P] :
( mem(V3P,arr(A_27b,bool))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V4x] : ap(f1870(A_27b,A_27a,V3P,V0f),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(V3P,ap(V0f,V4x))) ) ) ).
fof(lameq_f1871,axiom,
! [A_27b,V3P] :
( mem(V3P,arr(A_27b,bool))
=> ! [V5x] : ap(f1871(A_27b,V3P),V5x) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V5x),ap(V3P,V5x)) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__IMAGE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1y] :
( mem(V1y,A_27b)
=> ? [V2x] :
( mem(V2x,A_27a)
& ap(V0f,V2x) = V1y ) )
<=> ! [V3P] :
( 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),f1870(A_27b,A_27a,V3P,V0f))) = ap(c_2Epred__set_2EGSPEC(A_27b,A_27b),f1871(A_27b,V3P)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__ON__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ! [V2v] :
( mem(V2v,arr(A_27b,bool))
=> ( ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),V3t),V2v))
=> ? [V4s] :
( 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] :
( mem(V5y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V5y),V2v))
=> ? [V6x] :
( mem(V6x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1u))
& ap(V0f,V6x) = V5y ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ? [V2s] :
( mem(V2s,arr(A_27a,bool))
& ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s) = V1t ) )
<=> ! [V3y] :
( mem(V3y,A_27b)
=> ? [V4x] :
( mem(V4x,A_27a)
& ap(V0f,V4x) = V3y ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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] :
( 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] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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 ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__IMAGE__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( 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)) )
=> 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) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__IMAGE__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE__GEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( 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))
& ap(c_2Epred__set_2ECARD(A_27a),V0s) = 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] :
( mem(V3y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1t))
=> ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
& ap(V2f,V4x) = V3y ) ) )
<=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V6y] :
( 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 ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESURJECTIVE__IFF__INJECTIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( 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] :
( mem(V2y,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V0s))
=> ? [V3x] :
( mem(V3x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
& ap(V1f,V3x) = V2y ) ) )
<=> ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__EQ__BIJECTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1t))
& ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Epred__set_2ECARD(A_27b),V1t) )
=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& ! [V3x] :
( 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] :
( mem(V4y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t))
=> ? [V5x] :
( mem(V5x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s))
& ap(V2f,V5x) = V4y ) ) )
& ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7y] :
( 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 ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__EQ__BIJECTIONS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1t))
& ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Epred__set_2ECARD(A_27b),V1t) )
=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& ! [V4x] :
( 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] :
( 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 ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ESING__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINJECTIVE__ON__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1u] :
( mem(V1u,arr(A_27a,bool))
=> ( ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( 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] :
( mem(V4x,A_27a)
=> ! [V5y] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINJECTIVE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ap(V0f,V3x) = ap(V0f,V4y)
=> V3x = V4y ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__FINITE__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Enum__FINITE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
<=> ? [V1a] :
( mem(V1a,ty_2Enum_2Enum)
& ! [V2x] :
( mem(V2x,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2x),V0s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V2x),V1a)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Enum__FINITE__AVOID,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ? [V1a] :
( mem(V1a,ty_2Enum_2Enum)
& ~ p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1a),V0s)) ) ) ) ).
fof(conj_thm_2Ecardinal_2Enum__INFINITE,axiom,
~ p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ).
fof(ax_thm_2Ecardinal_2EHAS__SIZE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V1n))
<=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& ap(c_2Epred__set_2ECARD(A_27a),V0s) = V1n ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V1n))
=> ap(c_2Epred__set_2ECARD(A_27a),V0s) = V1n ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),c_2Enum_2E0))
<=> V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__SUC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),ap(c_2Enum_2ESUC,V1n)))
<=> ( V0s != c_2Epred__set_2EEMPTY(A_27a)
& ! [V2a] :
( 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)),V1n)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__HAS__SIZE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V1s),c_2Enum_2E0))
<=> V1s = c_2Epred__set_2EEMPTY(A_27a) )
& ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V1s),ap(c_2Enum_2ESUC,V0n)))
<=> ? [V2a] :
( mem(V2a,A_27a)
& ? [V3t] :
( mem(V3t,arr(A_27a,bool))
& p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V3t),V0n))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2a),V3t))
& V1s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V2a),V3t) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__SUBSET__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,arr(A_27a,bool))
=> ! [V1b] :
( 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))
& ap(c_2Epred__set_2ECARD(A_27a),V0a) = ap(c_2Epred__set_2ECARD(A_27a),V1b) )
=> V0a = V1b ) ) ) ) ).
fof(lameq_f1872,axiom,
! [A_27a,V4x] :
( mem(V4x,A_27a)
=> ! [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V5m] : ap(f1872(A_27a,V4x,V2f,V1n),V5m) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Eprim__rec_2E_3C,V5m),V1n)),ap(ap(c_2Emin_2E_3D(A_27a),ap(V2f,V5m)),V4x)) ) ) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__INDEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V1n))
=> ? [V2f] :
( mem(V2f,arr(ty_2Enum_2Enum,A_27a))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V3m),V1n))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V2f,V3m)),V0s)) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
=> p(ap(c_2Ebool_2E_3F_21(ty_2Enum_2Enum),f1872(A_27a,V4x,V2f,V1n))) ) ) ) ) ) ) ) ).
fof(lameq_f1873,axiom,
! [A_27b,A_27a,V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V5x] : ap(f1873(A_27b,A_27a,V1t,V0s),V5x) = ap(ap(c_2Epair_2E_2C(arr(A_27b,bool),bool),ap(V1t,V5x)),ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__BIGUNION__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,arr(A_27b,bool)))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),V0s),V2m))
& ! [V4x] :
( 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))),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),f1873(A_27b,A_27a,V1t,V0s))))),ap(ap(c_2Earithmetic_2E_2A,V2m),V3n))) ) ) ) ) ) ) ) ).
fof(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,c_2Earithmetic_2EZERO)))) ).
fof(conj_thm_2Ecardinal_2ECARD__BOOL,axiom,
ap(c_2Epred__set_2ECARD(bool),c_2Epred__set_2EUNIV(bool)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)) ).
fof(conj_thm_2Ecardinal_2EFINITE__BOOL,axiom,
p(ap(c_2Epred__set_2EFINITE(bool),c_2Epred__set_2EUNIV(bool))) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__CART__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0m))
=> p(ap(ap(c_2Ecardinal_2EHAS__SIZE(A_27a),c_2Epred__set_2EUNIV(A_27a)),ap(ap(c_2Earithmetic_2EEXP,V0m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__CART__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a)))
=> ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EUNIV(A_27a)) = 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,c_2Earithmetic_2EZERO))) ) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__CART__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ( 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))) ) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LT,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f275(V0n))),V0n)) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__NUMSEG__LT,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f275(V0n)))) ) ).
fof(lameq_f1874,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] : ap(f1874(V0n),V1m) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V1m),ap(ap(c_2Earithmetic_2E_3C_3D,V1m),V0n)) ) ).
fof(conj_thm_2Ecardinal_2EHAS__SIZE__NUMSEG__LE,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ecardinal_2EHAS__SIZE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1874(V0n))),ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))))) ) ).
fof(conj_thm_2Ecardinal_2EFINITE__NUMSEG__LE,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f1874(V0n)))) ) ).
fof(conj_thm_2Ecardinal_2EINFINITE__DIFF__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELE__CASES,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n))
| p(ap(ap(c_2Earithmetic_2E_3C_3D,V1n),V0m)) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELT__CASES,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n))
| p(ap(ap(c_2Eprim__rec_2E_3C,V1n),V0m))
| V0m = V1n ) ) ) ).
fof(conj_thm_2Ecardinal_2ELT,axiom,
( ! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),c_2Enum_2E0))
<=> $false ) )
& ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V1m),ap(c_2Enum_2ESUC,V2n)))
<=> ( V1m = V2n
| p(ap(ap(c_2Eprim__rec_2E_3C,V1m),V2n)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELT__LE,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n))
<=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n))
& V0m != V1n ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EGE,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3E_3D,V1m),V0n))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0n),V1m)) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELE__SUC__LT,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Enum_2ESUC,V0m)),V1n))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINJECTIVE__ON__LEFT__INVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( 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] :
( mem(V4g,arr(A_27b,A_27a))
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V1s))
=> ap(V4g,ap(V0f,V5x)) = V5x ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINJECTIVE__LEFT__INVERSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ap(V0f,V1x) = ap(V0f,V2y)
=> V1x = V2y ) ) )
<=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& ! [V4x] :
( mem(V4x,A_27a)
=> ap(V3g,ap(V0f,V4x)) = V4x ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EINTER__ACI,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(A_27a,bool))
=> ! [V2q] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECONJ__ACI,axiom,
! [V0r] :
( mem(V0r,bool)
=> ! [V1p] :
( mem(V1p,bool)
=> ! [V2q] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EUNION__ACI,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(A_27a,bool))
=> ! [V1p] :
( mem(V1p,arr(A_27a,bool))
=> ! [V2q] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELT__NZ,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n))
<=> V0n != c_2Enum_2E0 ) ) ).
fof(conj_thm_2Ecardinal_2ELE__1,axiom,
( ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( V0n != c_2Enum_2E0
=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n)) ) )
& ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( V1n != c_2Enum_2E0
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V1n)) ) )
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V2n))
=> V2n != c_2Enum_2E0 ) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V3n))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V3n)) ) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V4n))
=> p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V4n)) ) )
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),V5n))
=> V5n != c_2Enum_2E0 ) ) ) ).
fof(conj_thm_2Ecardinal_2EOR__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
| ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1Q,V3x)) ) )
<=> ? [V4x] :
( mem(V4x,A_27a)
& ( p(ap(V0P,V4x))
| p(ap(V1Q,V4x)) ) ) ) ) ) ) ).
fof(lameq_f1875,axiom,
! [A_27b,A_27a,V4y] :
( mem(V4y,A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V5x] : ap(f1875(A_27b,A_27a,V4y,V0f,V1s),V5x) = 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)) ) ) ) ).
fof(conj_thm_2Ecardinal_2EBIJECTIVE__INJECTIVE__SURJECTIVE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( ! [V3x] :
( 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] :
( mem(V4y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f1875(A_27b,A_27a,V4y,V0f,V1s))) ) ) )
<=> ( ! [V6x] :
( 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] :
( mem(V7x,A_27a)
=> ! [V8y] :
( 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] :
( mem(V9y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V9y),V2t))
=> ? [V10x] :
( mem(V10x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V10x),V1s))
& ap(V0f,V10x) = V9y ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EBIJECTIVE__INVERSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( ( ! [V3x] :
( 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] :
( mem(V4y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f1875(A_27b,A_27a,V4y,V0f,V1s))) ) ) )
<=> ( ! [V6x] :
( 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] :
( mem(V7g,arr(A_27b,A_27a))
& ! [V8y] :
( 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] :
( mem(V9y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V9y),V2t))
=> ap(V0f,ap(V7g,V9y)) = V9y ) )
& ! [V10x] :
( mem(V10x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V10x),V1s))
=> ap(V7g,ap(V0f,V10x)) = V10x ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ele__c,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& ! [V3x] :
( 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] :
( mem(V4x,A_27a)
=> ! [V5y] :
( 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 ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Elt__c,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ) ).
fof(lameq_f1876,axiom,
! [A_27b,A_27a,V4y] :
( mem(V4y,A_27b)
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V5x] : ap(f1876(A_27b,A_27a,V4y,V2f,V0s),V5x) = 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)) ) ) ) ).
fof(conj_thm_2Ecardinal_2Eeq__c,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& ! [V3x] :
( 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] :
( mem(V4y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f1876(A_27b,A_27a,V4y,V2f,V0s))) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Ecardinal_2Ecardgeq__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Ege__c,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(ax_thm_2Ecardinal_2Ecardgt__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2Egt__c,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ELE__C,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27b),V0s),V1t))
<=> ? [V2g] :
( mem(V2g,arr(A_27b,A_27a))
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ? [V4y] :
( mem(V4y,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V1t))
& ap(V2g,V4y) = V3x ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2EGE__C,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardgeq(A_27a,A_27b),V0s),V1t))
<=> ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& ! [V3y] :
( mem(V3y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V1t))
=> ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
& V3y = ap(V2f,V4x) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECOUNTABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( 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)) ) ) ) ).
fof(lameq_f1877,axiom,
! [A_27a,A_27b,V2R] :
( mem(V2R,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V6y] : ap(f1877(A_27a,A_27b,V2R,V5x,V1t),V6y) = 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))) ) ) ) ).
fof(lameq_f1878,axiom,
! [A_27b,A_27a,V2R] :
( mem(V2R,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ! [V7y] :
( mem(V7y,A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V8x] : ap(f1878(A_27b,A_27a,V2R,V7y,V0s),V8x) = 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))) ) ) ) ).
fof(conj_thm_2Ecardinal_2EEQ__C,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27b),V0s),V1t))
<=> ? [V2R] :
( mem(V2R,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V0s))
=> p(ap(c_2Ebool_2E_3F_21(A_27b),f1877(A_27a,A_27b,V2R,V5x,V1t))) ) )
& ! [V7y] :
( mem(V7y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V7y),V1t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f1878(A_27b,A_27a,V2R,V7y,V0s))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LT__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ~ ~ p(ap(ap(c_2Ecardinal_2Ecardleq(A_27a,A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LET__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LTE__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LT__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__EQ__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Ecardinal_2Ecardeq(A_27a,A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__EQ__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__EQ__IMP__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LT__IMP__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(lameq_f1879,axiom,
! [A_27b,A_27a,V5y] :
( mem(V5y,A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V6x] : ap(f1879(A_27b,A_27a,V5y,V0R,V1s),V6x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1s)),ap(ap(V0R,V6x),V5y)) ) ) ) ).
fof(lameq_f1880,axiom,
! [A_27a,A_27b,V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V5y] : ap(f1880(A_27a,A_27b,V0R,V1s),V5y) = ap(ap(c_2Epair_2E_2C(A_27b,bool),V5y),ap(c_2Ebool_2E_3F(A_27a),f1879(A_27b,A_27a,V5y,V0R,V1s))) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__RELATIONAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V4y_27] :
( 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),f1880(A_27a,A_27b,V0R,V1s))),V1s)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( 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) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__EQ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( 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) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__ANTISYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LE__TOTAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LET__TOTAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LTE__TOTAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__LT__TOTAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Ecardinal_2ECARD__NOT__LE,conjecture,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
%------------------------------------------------------------------------------