TPTP Problem File: ITP005^5.p
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%------------------------------------------------------------------------------
% File : ITP005^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Eset__relation_2Erel__to__reln__inv.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_2Eset__relation_2Erel__to__reln__inv.p [Gau20]
% : HL402001^5.p [TPAP]
% Status : Theorem
% Rating : 0.50 v9.0.0, 0.60 v8.2.0, 0.54 v8.1.0, 0.55 v7.5.0
% Syntax : Number of formulae : 3387 ( 335 unt; 494 typ; 0 def)
% Number of atoms : 43012 (2090 equ; 0 cnn)
% Maximal formula atoms : 110 ( 14 avg)
% Number of connectives : 86778 ( 369 ~; 262 |;1678 &;78007 @)
% ( 968 <=>;5494 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 11 avg)
% Number of types : 10 ( 8 usr)
% Number of type conns : 514 ( 514 >; 0 *; 0 +; 0 <<)
% Number of symbols : 525 ( 522 usr; 170 con; 0-4 aty)
% Number of variables : 9302 ( 433 ^;8552 !; 317 ?;9302 :)
% SPC : TH0_THM_EQU_NAR
% 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').
%------------------------------------------------------------------------------
thf(tp_c_2Eset__relation_2ERREFL__EXP,type,
c_2Eset__relation_2ERREFL__EXP: del > $i ).
thf(mem_c_2Eset__relation_2ERREFL__EXP,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Eset__relation_2ERRUNIV,type,
c_2Eset__relation_2ERRUNIV: del > $i ).
thf(mem_c_2Eset__relation_2ERRUNIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Eacyclic,type,
c_2Eset__relation_2Eacyclic: del > $i ).
thf(mem_c_2Eset__relation_2Eacyclic,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eset__relation_2Eall__choices,type,
c_2Eset__relation_2Eall__choices: del > $i ).
thf(mem_c_2Eset__relation_2Eall__choices,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eall__choices @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eantisym,type,
c_2Eset__relation_2Eantisym: del > $i ).
thf(mem_c_2Eset__relation_2Eantisym,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eset__relation_2Echain,type,
c_2Eset__relation_2Echain: del > $i ).
thf(mem_c_2Eset__relation_2Echain,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Echain @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Edomain,type,
c_2Eset__relation_2Edomain: del > del > $i ).
thf(mem_c_2Eset__relation_2Edomain,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Efchains,type,
c_2Eset__relation_2Efchains: del > $i ).
thf(mem_c_2Eset__relation_2Efchains,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Efchains @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Efinite__prefixes,type,
c_2Eset__relation_2Efinite__prefixes: del > del > $i ).
thf(mem_c_2Eset__relation_2Efinite__prefixes,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eget__min,type,
c_2Eset__relation_2Eget__min: del > $i ).
thf(mem_c_2Eset__relation_2Eget__min,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eget__min @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Eset__relation_2Eirreflexive,type,
c_2Eset__relation_2Eirreflexive: del > $i ).
thf(mem_c_2Eset__relation_2Eirreflexive,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Elinear__order,type,
c_2Eset__relation_2Elinear__order: del > $i ).
thf(mem_c_2Eset__relation_2Elinear__order,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Emaximal__elements,type,
c_2Eset__relation_2Emaximal__elements: del > $i ).
thf(mem_c_2Eset__relation_2Emaximal__elements,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Emaximal__elements @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Eminimal__elements,type,
c_2Eset__relation_2Eminimal__elements: del > $i ).
thf(mem_c_2Eset__relation_2Eminimal__elements,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eminimal__elements @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Enth__min,type,
c_2Eset__relation_2Enth__min: del > $i ).
thf(mem_c_2Eset__relation_2Enth__min,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Enth__min @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).
thf(tp_c_2Eset__relation_2Enum__order,type,
c_2Eset__relation_2Enum__order: del > $i ).
thf(mem_c_2Eset__relation_2Enum__order,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Enum__order @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Epartial__order,type,
c_2Eset__relation_2Epartial__order: del > $i ).
thf(mem_c_2Eset__relation_2Epartial__order,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eper,type,
c_2Eset__relation_2Eper: del > $i ).
thf(mem_c_2Eset__relation_2Eper,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eper @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eper__restrict,type,
c_2Eset__relation_2Eper__restrict: del > $i ).
thf(mem_c_2Eset__relation_2Eper__restrict,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Eper__restrict @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Erange,type,
c_2Eset__relation_2Erange: del > del > $i ).
thf(mem_c_2Eset__relation_2Erange,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Ercomp,type,
c_2Eset__relation_2Ercomp: del > del > del > $i ).
thf(mem_c_2Eset__relation_2Ercomp,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Ereflexive,type,
c_2Eset__relation_2Ereflexive: del > $i ).
thf(mem_c_2Eset__relation_2Ereflexive,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Erel__to__reln,type,
c_2Eset__relation_2Erel__to__reln: del > del > $i ).
thf(mem_c_2Eset__relation_2Erel__to__reln,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Ereln__to__rel,type,
c_2Eset__relation_2Ereln__to__rel: del > del > $i ).
thf(mem_c_2Eset__relation_2Ereln__to__rel,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Errestrict,type,
c_2Eset__relation_2Errestrict: del > $i ).
thf(mem_c_2Eset__relation_2Errestrict,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Eset__relation_2Estrict,type,
c_2Eset__relation_2Estrict: del > $i ).
thf(mem_c_2Eset__relation_2Estrict,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Estrict__linear__order,type,
c_2Eset__relation_2Estrict__linear__order: del > $i ).
thf(mem_c_2Eset__relation_2Estrict__linear__order,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Etc,type,
c_2Eset__relation_2Etc: del > $i ).
thf(mem_c_2Eset__relation_2Etc,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Etransitive,type,
c_2Eset__relation_2Etransitive: del > $i ).
thf(mem_c_2Eset__relation_2Etransitive,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Eset__relation_2Euniv__reln,type,
c_2Eset__relation_2Euniv__reln: del > $i ).
thf(mem_c_2Eset__relation_2Euniv__reln,axiom,
! [A_27a: del] : ( mem @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eset__relation_2Eupper__bounds,type,
c_2Eset__relation_2Eupper__bounds: del > del > $i ).
thf(mem_c_2Eset__relation_2Eupper__bounds,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erextension,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( V0s = V1t )
<=> ! [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 ) ) @ V0s ) )
<=> ( 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 ) ) @ V1t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Edomain__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V0r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1x )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Erange__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V0r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1y )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V2x ) @ V1y ) ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__domain,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V1r ) ) )
<=> ? [V2y: $i] :
( ( mem @ V2y @ 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 ) @ V0x ) @ V2y ) ) @ V1r ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__range,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0y ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ V1r ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V2x ) @ V0y ) ) @ V1r ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__dom__rg,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) ) @ V2r ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V2r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V1y ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V2r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Edomain__mono,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0r ) @ V1r_27 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ V1r_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erange__mono,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0r ) @ V1r_27 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V1r_27 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Errestrict__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__rrestrict,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V2r ) @ V3s ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ V2r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V3s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1y ) @ V3s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ein__rrestrict__alt,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0x ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r ) @ V2s ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0x ) @ V1r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27a ) @ V0x ) ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27a ) @ V0x ) ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) @ V0r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__union,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V1r2 ) ) @ V2s )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r1 ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V1r2 ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1x ) ) @ V2y )
= ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Edomain__rrestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erange__rrestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) @ V1s ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ercomp__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c ) @ V0r1 ) @ V1r2 )
= ( 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 @ ( c_2Ebool_2E_3F @ A_27c )
@ ( lam @ A_27c
@ ^ [V4z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ V4z ) ) @ V0r1 ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27b ) @ V4z ) @ V3y ) ) @ V1r2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Estrict__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) @ V1s ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Euniv__reln__def,axiom,
! [A_27a: del,V0xs: $i] :
( ( mem @ V0xs @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Euniv__reln @ A_27a ) @ V0xs )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V1x1: $i] :
( lam @ A_27a
@ ^ [V2x2: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x1 ) @ V2x2 ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x1 ) @ V0xs ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x2 ) @ V0xs ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Efinite__prefixes__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
<=> ! [V2e: $i] :
( ( mem @ V2e @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2e ) @ V1s ) )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3e_27: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3e_27 ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3e_27 ) @ V2e ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__s,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V2s_27 ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V2s_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__r,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V1r_27 ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V1r_27 ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset__rs,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2r_27: $i] :
( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2r_27 ) @ V0r ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V3s_27 ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V2r_27 ) @ V3s_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__subset,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V0r ) @ V2s_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V2s_27 ) ) @ V2s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__union,axiom,
! [A_27a: del,A_27b: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V2s1: $i] :
( ( mem @ V2s1 @ ( arr @ A_27b @ bool ) )
=> ! [V3s2: $i] :
( ( mem @ V3s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r1 ) @ V2s1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V1r2 ) @ V3s2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0r1 ) @ V1r2 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V2s1 ) @ V3s2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__comp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) @ bool ) )
=> ! [V2s1: $i] :
( ( mem @ V2s1 @ ( arr @ A_27b @ bool ) )
=> ! [V3s2: $i] :
( ( mem @ V3s2 @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r1 ) @ V2s1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27b @ A_27c ) @ V1r2 ) @ V3s2 ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V4x: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V4x )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27c )
@ ( lam @ A_27c
@ ^ [V5y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27c ) @ V5y ) @ V3s2 ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ V4x ) @ V5y ) ) @ V1r2 ) ) ) ) ) ) ) )
@ V2s1 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27c @ A_27b ) @ V0r1 ) @ V1r2 ) ) @ V3s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__inj__image,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) )
=> ( V3x = V4y ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a ) @ V1r ) @ V2s ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27b @ A_27b )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V5x: $i] :
( lam @ A_27a
@ ^ [V6y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V0f @ V5x ) ) @ ( ap @ V0f @ V6y ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5x ) @ V6y ) ) @ V1r ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Efinite__prefixes__range,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ V2t ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a ) @ V0r ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Etc__def,axiom,
! [A_27a: del] :
( ( c_2Eset__relation_2Etc @ A_27a )
= ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool )
@ ^ [V0r: $i] :
( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V1a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool )
@ ^ [V2tc_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V3a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V3a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) ) @ ( ap @ V0r @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V7y: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V3a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V7y ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V8z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2tc_27 @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V8z ) ) ) @ ( ap @ V2tc_27 @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V8z ) @ V7y ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2tc_27 @ V3a0 ) ) ) ) )
@ ( ap @ V2tc_27 @ V1a0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__rules,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ? [V5z: $i] :
( ( mem @ V5z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V5z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5z ) @ V4y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V4y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__cases,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
| ? [V3z: $i] :
( ( mem @ V3z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3z ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__ind,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__closure,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Esubset__tc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__idemp,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) )
= ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__mono,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__strongind,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__cases__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
| ? [V3z: $i] :
( ( mem @ V3z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3z ) @ V2y ) ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__cases__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
| ? [V3z: $i] :
( ( mem @ V3z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3z ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__ind__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__strongind__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V6z ) @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__ind__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ertc__ind__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ? [V5z: $i] :
( ( mem @ V5z @ A_27a )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V3x ) @ V5z ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5z ) @ V4y ) ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V3x ) @ V4y ) ) ) ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V7y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V6x ) @ V7y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__strongind__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1tc_27: $i] :
( ( mem @ V1tc_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ? [V6z: $i] :
( ( mem @ V6z @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V6z ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V6z ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6z ) @ V5y ) ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V7x ) @ V8y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ V1tc_27 @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__union,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r1 ) ) )
=> ! [V3r2: $i] :
( ( mem @ V3r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V3r2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__implication,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r1 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V1r2 ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r1 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V1r2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__empty,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__empty__eqn,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__domain__range,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Errestrict__tc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ! [V3e_27: $i] :
( ( mem @ V3e_27 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2e ) @ V3e_27 ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1x ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2e ) @ V3e_27 ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__SWAP,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ V0r ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eacyclic__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__subset,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1r2 ) @ V0r1 ) ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__irreflexive,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__SWAP,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) ) @ V0r ) ) )
<=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__bigunion,axiom,
! [A_27a: del,V0rs: $i] :
( ( mem @ V0rs @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) )
=> ( ( ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V2r_27: $i] :
( ( mem @ V2r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V1r ) @ V0rs ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V2r_27 ) @ V0rs ) )
& ( V1r != V2r_27 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V2r_27 ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V2r_27 ) ) ) ) ) ) )
& ! [V3r: $i] :
( ( mem @ V3r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) @ V3r ) @ V0rs ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V3r ) ) ) ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0rs ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eacyclic__union,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r_27: $i] :
( ( mem @ V1r_27 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V1r_27 ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V1r_27 ) ) ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ V1r_27 ) ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r ) @ V1r_27 ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ereflexive__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eirreflexive__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Eirreflexive @ A_27a ) @ V0r ) @ V1s ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Etransitive__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V3z ) ) @ V0r ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V3z ) ) @ V0r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Etransitive__tc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
=> ( ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r )
= V0r ) ) ) ).
thf(conj_thm_2Eset__relation_2Etc__transitive,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Etc @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Eantisym__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) @ V0r ) ) )
=> ( V1x = V2y ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Epartial__order__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Eantisym__subset,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1s ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__dom__rng,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [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 @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__subset,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V2s_27 ) ) @ V2s_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__partial__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) @ V1s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__partial__order__acyclic,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eacyclic @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Elinear__order__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
& ! [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 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__subset,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s_27 ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V2s_27 ) ) @ V2s_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Epartial__order__linear__order,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Epartial__order @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Estrict__linear__order__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) @ V1s ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V2x ) ) @ V0r ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V1s ) )
& ( V3x != V4y ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V4y ) ) @ V0r ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4y ) @ V3x ) ) @ V0r ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Estrict__linear__order__dom__rng,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [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 @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Estrict__linear__order @ A_27a ) @ V0r ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Elinear__order__dom__rng,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [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 @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Ereln__to__rel__def,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r )
= ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2Erel__to__reln__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R )
= ( 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
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $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 ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2ERRUNIV__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2ERRUNIV @ A_27a ) @ V0s )
= ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V0s ) ) ) ) ) ) ).
thf(ax_thm_2Eset__relation_2ERREFL__EXP__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V0R ) @ V1s )
= ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27a ) @ V0R )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2ERREFL__EXP__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2ERREFL__EXP__UNIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0R ) ) ).
thf(conj_thm_2Eset__relation_2EREL__RESTRICT__UNIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0R ) ) ).
thf(conj_thm_2Eset__relation_2Ein__rel__to__reln,axiom,
! [A_27a: del,A_27b: del,V0xy: $i] :
( ( mem @ V0xy @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0xy ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V1R ) ) )
<=> ( p @ ( ap @ ( ap @ V1R @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0xy ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0xy ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__to__rel__app,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ V0r ) @ V1x ) @ V2y ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ V0r ) ) ) ) ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__IS__UNCURRY,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) ) ).
thf(conj_thm_2Eset__relation_2Ereln__to__rel__IS__CURRY,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b )
= ( c_2Epair_2ECURRY @ A_27a @ A_27b @ bool ) ) ).
thf(conj_thm_2Eset__relation_2Erel__to__reln__inv,conjecture,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b ) @ ( ap @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b ) @ V0R ) )
= V0R ) ) ).
%------------------------------------------------------------------------------