TPTP Problem File: SET162+3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : SET162+3 : TPTP v9.0.0. Released v2.2.0.
% Domain : Set Theory
% Problem : The union of X and the empty set is X
% Version : [Try90] axioms : Reduced > Incomplete.
% English :
% Refs : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
% : [Try90] Trybulec (1990), Tarski Grothendieck Set Theory
% : [TS89] Trybulec & Swieczkowska (1989), Boolean Properties of
% Source : [ILF]
% Names : BOOLE (60) [TS89]
% Status : Theorem
% Rating : 0.09 v9.0.0, 0.11 v8.2.0, 0.08 v8.1.0, 0.03 v7.2.0, 0.00 v7.1.0, 0.04 v7.0.0, 0.00 v6.4.0, 0.04 v6.3.0, 0.12 v6.1.0, 0.17 v5.5.0, 0.15 v5.4.0, 0.18 v5.3.0, 0.26 v5.2.0, 0.05 v5.0.0, 0.08 v4.1.0, 0.13 v4.0.1, 0.17 v3.7.0, 0.10 v3.5.0, 0.11 v3.3.0, 0.07 v3.2.0, 0.18 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1
% Syntax : Number of formulae : 9 ( 4 unt; 0 def)
% Number of atoms : 18 ( 4 equ)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 11 ( 2 ~; 1 |; 1 &)
% ( 6 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 2 ( 2 usr; 1 con; 0-2 aty)
% Number of variables : 18 ( 18 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(boole - df(2),1833042)
fof(union_defn,axiom,
! [B,C,D] :
( member(D,union(B,C))
<=> ( member(D,B)
| member(D,C) ) ) ).
%---- line(hidden - axiom93,1832636)
fof(empty_set_defn,axiom,
! [B] : ~ member(B,empty_set) ).
%---- line(boole - df(8),1833103)
fof(equal_defn,axiom,
! [B,C] :
( B = C
<=> ( subset(B,C)
& subset(C,B) ) ) ).
%---- property(commutativity,op(union,2,function))
fof(commutativity_of_union,axiom,
! [B,C] : union(B,C) = union(C,B) ).
%---- line(tarski - df(3),1832749)
fof(subset_defn,axiom,
! [B,C] :
( subset(B,C)
<=> ! [D] :
( member(D,B)
=> member(D,C) ) ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(reflexivity_of_subset,axiom,
! [B] : subset(B,B) ).
%---- line(hidden - axiom95,1832628)
fof(empty_defn,axiom,
! [B] :
( empty(B)
<=> ! [C] : ~ member(C,B) ) ).
%---- line(hidden - axiom96,1832615)
fof(equal_member_defn,axiom,
! [B,C] :
( B = C
<=> ! [D] :
( member(D,B)
<=> member(D,C) ) ) ).
%---- line(boole - th(60),1833665)
fof(prove_union_empty_set,conjecture,
! [B] : union(B,empty_set) = B ).
%--------------------------------------------------------------------------