TPTP Problem File: SET662+3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : SET662+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : The empty set is a relation from X to Y
% 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 (25) [Wor90]
% Status : Theorem
% Rating : 0.25 v8.2.0, 0.19 v8.1.0, 0.25 v7.5.0, 0.22 v7.4.0, 0.27 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.26 v7.0.0, 0.23 v6.4.0, 0.27 v6.3.0, 0.25 v6.2.0, 0.24 v6.1.0, 0.27 v6.0.0, 0.35 v5.5.0, 0.26 v5.4.0, 0.21 v5.3.0, 0.22 v5.2.0, 0.15 v5.1.0, 0.14 v5.0.0, 0.12 v4.1.0, 0.13 v4.0.0, 0.12 v3.7.0, 0.15 v3.5.0, 0.16 v3.4.0, 0.11 v3.3.0, 0.14 v3.2.0, 0.18 v3.1.0, 0.11 v2.7.0, 0.17 v2.6.0, 0.14 v2.5.0, 0.12 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1
% Syntax : Number of formulae : 22 ( 3 unt; 0 def)
% Number of atoms : 75 ( 1 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 58 ( 5 ~; 0 |; 7 &)
% ( 6 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 8 ( 8 usr; 2 con; 0-2 aty)
% Number of variables : 43 ( 38 !; 5 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(relat_1 - th(58),1918738)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> subset(empty_set,cross_product(B,C)) ) ) ).
%---- line(relset_1 - df(1),1916080)
fof(p2,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(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ? [D] : ilf_type(D,relation_type(C,B)) ) ) ).
%---- line(hidden - axiom233,1832636)
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ~ member(B,empty_set) ) ).
%---- declaration(line(hidden - axiom233,1832636)) Part 1
fof(p5a,axiom,
empty(empty_set) ).
%---- declaration(line(hidden - axiom233,1832636)) Part 2
fof(p5b,axiom,
type(empty_set,set_type) ).
%---- declaration(op(cross_product,2,function))
fof(p6,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),set_type) ) ) ).
%---- line(hidden - axiom234,1832648)
fof(p7,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 - axiom234,1832648))
fof(p8,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- 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) ) ) ) ) ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(p10,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom236,1832628)
fof(p11,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- line(hidden - axiom238,1832644)
fof(p12,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 - axiom238,1832644))
fof(p13,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom239,1832640)
fof(p14,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 - axiom239,1832640))
fof(p15,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p16,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(axiom240,relation_like)
fof(p17,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- conditional_cluster(axiom241,relation_like)
fof(p18,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) ) ) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p19,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- declaration(set)
fof(p20,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(25),1916495)
fof(prove_relset_1_25,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(empty_set,relation_type(B,C)) ) ) ).
%--------------------------------------------------------------------------