TPTP Problem File: SET665+3.p
View Solutions
- Solve Problem
%--------------------------------------------------------------------------
% File : SET665+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : The identity relation on X is a subset of X x X
% Version : [Wor90] axioms : Reduced > Incomplete.
% English :
% Refs : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
% : [Wor90] Woronowicz (1990), Relations Defined on Sets
% Source : [ILF]
% Names : RELSET_1 (28) [Wor90]
% Status : Theorem
% Rating : 0.44 v8.2.0, 0.53 v7.5.0, 0.50 v7.4.0, 0.37 v7.3.0, 0.52 v7.2.0, 0.48 v7.0.0, 0.43 v6.4.0, 0.58 v6.3.0, 0.54 v6.2.0, 0.56 v6.1.0, 0.63 v6.0.0, 0.57 v5.5.0, 0.74 v5.4.0, 0.75 v5.3.0, 0.81 v5.2.0, 0.65 v5.1.0, 0.67 v5.0.0, 0.71 v4.1.0, 0.65 v4.0.1, 0.70 v4.0.0, 0.62 v3.7.0, 0.70 v3.5.0, 0.63 v3.4.0, 0.58 v3.3.0, 0.57 v3.2.0, 0.64 v3.1.0, 0.67 v2.6.0, 0.86 v2.5.0, 0.75 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax : Number of formulae : 28 ( 2 unt; 0 def)
% Number of atoms : 114 ( 4 equ)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 90 ( 4 ~; 0 |; 15 &)
% ( 12 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 9 ( 9 usr; 2 con; 0-2 aty)
% Number of variables : 64 ( 56 !; 8 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(relat_1 - df(3),1917829)
fof(p1,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ( subset(B,C)
<=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,set_type)
=> ( member(ordered_pair(D,E),B)
=> member(ordered_pair(D,E),C) ) ) ) ) ) ) ).
%---- line(relat_1 - th(69),1918879)
fof(p2,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( member(ordered_pair(C,D),identity_relation_of(B))
<=> ( member(C,B)
& C = D ) ) ) ) ) ).
%---- line(zfmisc_1 - th(106),1905180)
fof(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,set_type)
=> ( member(ordered_pair(B,C),cross_product(D,E))
<=> ( member(B,D)
& member(C,E) ) ) ) ) ) ) ).
%---- line(relset_1 - th(5),1916109)
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),relation_type(B,C)) ) ) ).
%---- line(zfmisc_1 - df(1),1903822)
fof(p5,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,cross_product(B,C))
<=> ? [E] :
( ilf_type(E,set_type)
& ? [F] :
( ilf_type(F,set_type)
& member(E,B)
& member(F,C)
& D = ordered_pair(E,F) ) ) ) ) ) ) ).
%---- declaration(line(zfmisc_1 - df(1),1903822))
fof(p6,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),set_type) ) ) ).
%---- line(relat_1 - df(10),1918876)
fof(p7,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( member(ordered_pair(C,D),identity_relation_of(B))
<=> ( member(C,B)
& C = D ) ) ) ) ) ).
%---- declaration(line(relat_1 - df(10),1918876))
fof(p8,axiom,
! [B] :
( ilf_type(B,set_type)
=> ilf_type(identity_relation_of(B),binary_relation_type) ) ).
%---- line(tarski - df(3),1832749)
fof(p9,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( subset(B,C)
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p10,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- line(relat_1 - axiom283,1917641)
fof(p11,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(B,binary_relation_type)
<=> ( relation_like(B)
& ilf_type(B,set_type) ) ) ) ).
%---- type_nonempty(line(relat_1 - axiom283,1917641))
fof(p12,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(relset_1 - df(1),1916080)
fof(p13,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> ilf_type(D,relation_type(B,C)) )
& ! [E] :
( ilf_type(E,relation_type(B,C))
=> ilf_type(E,subset_type(cross_product(B,C))) ) ) ) ) ).
%---- type_nonempty(line(relset_1 - df(1),1916080))
fof(p14,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ? [D] : ilf_type(D,relation_type(C,B)) ) ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(p15,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(p16,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom285,1832648)
fof(p17,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ilf_type(C,subset_type(B))
<=> ilf_type(C,member_type(power_set(B))) ) ) ) ).
%---- type_nonempty(line(hidden - axiom285,1832648))
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p19,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( relation_like(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ( member(C,B)
=> ? [D] :
( ilf_type(D,set_type)
& ? [E] :
( ilf_type(E,set_type)
& C = ordered_pair(D,E) ) ) ) ) ) ) ).
%---- conditional_cluster(axiom287,relation_like)
fof(p20,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> relation_like(D) ) ) ) ).
%---- line(hidden - axiom288,1832644)
fof(p21,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(B,power_set(C))
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ) ).
%---- declaration(line(hidden - axiom288,1832644))
fof(p22,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom289,1832640)
fof(p23,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ( ~ empty(C)
& ilf_type(C,set_type) )
=> ( ilf_type(B,member_type(C))
<=> member(B,C) ) ) ) ).
%---- type_nonempty(line(hidden - axiom289,1832640))
fof(p24,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(hidden - axiom290,1832628)
fof(p25,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom291,empty)
fof(p26,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- declaration(set)
fof(p27,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(28),1916537)
fof(prove_relset_1_28,conjecture,
! [B] :
( ilf_type(B,set_type)
=> subset(identity_relation_of(B),cross_product(B,B)) ) ).
%--------------------------------------------------------------------------