## TPTP Problem File: SEU140+2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU140+2 : TPTP v7.1.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP chainy problem t63_xboole_1
% Version  : [Urb07] axioms : Especial.
% English  :

% Refs     : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
%          : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb07]
% Names    : chainy-t63_xboole_1 [Urb07]

% Status   : Theorem
% Rating   : 0.10 v7.1.0, 0.09 v7.0.0, 0.10 v6.4.0, 0.15 v6.3.0, 0.21 v6.2.0, 0.24 v6.1.0, 0.20 v6.0.0, 0.22 v5.5.0, 0.26 v5.4.0, 0.29 v5.3.0, 0.26 v5.2.0, 0.10 v5.0.0, 0.17 v4.0.1, 0.22 v4.0.0, 0.25 v3.7.0, 0.20 v3.5.0, 0.21 v3.4.0, 0.26 v3.3.0
% Syntax   : Number of formulae    :   56 (  24 unit)
%            Number of atoms       :  109 (  27 equality)
%            Maximal formula depth :    9 (   4 average)
%            Number of connectives :   76 (  23 ~  ;   1  |;  20  &)
%                                         (  14 <=>;  18 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    7 (   1 propositional; 0-2 arity)
%            Number of functors    :    4 (   1 constant; 0-2 arity)
%            Number of variables   :  111 (   4 singleton; 107 !;   4 ?)
%            Maximal term depth    :    3 (   1 average)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,(
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) )).

fof(antisymmetry_r2_xboole_0,axiom,(
! [A,B] :
( proper_subset(A,B)
=> ~ proper_subset(B,A) ) )).

fof(commutativity_k2_xboole_0,axiom,(
! [A,B] : set_union2(A,B) = set_union2(B,A) )).

fof(commutativity_k3_xboole_0,axiom,(
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) )).

fof(d10_xboole_0,axiom,(
! [A,B] :
( A = B
<=> ( subset(A,B)
& subset(B,A) ) ) )).

fof(d1_xboole_0,axiom,(
! [A] :
( A = empty_set
<=> ! [B] : ~ in(B,A) ) )).

fof(d2_xboole_0,axiom,(
! [A,B,C] :
( C = set_union2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
| in(D,B) ) ) ) )).

fof(d3_tarski,axiom,(
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) )).

fof(d3_xboole_0,axiom,(
! [A,B,C] :
( C = set_intersection2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
& in(D,B) ) ) ) )).

fof(d4_xboole_0,axiom,(
! [A,B,C] :
( C = set_difference(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
& ~ in(D,B) ) ) ) )).

fof(d7_xboole_0,axiom,(
! [A,B] :
( disjoint(A,B)
<=> set_intersection2(A,B) = empty_set ) )).

fof(d8_xboole_0,axiom,(
! [A,B] :
( proper_subset(A,B)
<=> ( subset(A,B)
& A != B ) ) )).

fof(dt_k1_xboole_0,axiom,(
\$true )).

fof(dt_k2_xboole_0,axiom,(
\$true )).

fof(dt_k3_xboole_0,axiom,(
\$true )).

fof(dt_k4_xboole_0,axiom,(
\$true )).

fof(fc1_xboole_0,axiom,(
empty(empty_set) )).

fof(fc2_xboole_0,axiom,(
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) )).

fof(fc3_xboole_0,axiom,(
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) )).

fof(idempotence_k2_xboole_0,axiom,(
! [A,B] : set_union2(A,A) = A )).

fof(idempotence_k3_xboole_0,axiom,(
! [A,B] : set_intersection2(A,A) = A )).

fof(irreflexivity_r2_xboole_0,axiom,(
! [A,B] : ~ proper_subset(A,A) )).

fof(l32_xboole_1,lemma,(
! [A,B] :
( set_difference(A,B) = empty_set
<=> subset(A,B) ) )).

fof(rc1_xboole_0,axiom,(
? [A] : empty(A) )).

fof(rc2_xboole_0,axiom,(
? [A] : ~ empty(A) )).

fof(reflexivity_r1_tarski,axiom,(
! [A,B] : subset(A,A) )).

fof(symmetry_r1_xboole_0,axiom,(
! [A,B] :
( disjoint(A,B)
=> disjoint(B,A) ) )).

fof(t12_xboole_1,lemma,(
! [A,B] :
( subset(A,B)
=> set_union2(A,B) = B ) )).

fof(t17_xboole_1,lemma,(
! [A,B] : subset(set_intersection2(A,B),A) )).

fof(t19_xboole_1,lemma,(
! [A,B,C] :
( ( subset(A,B)
& subset(A,C) )
=> subset(A,set_intersection2(B,C)) ) )).

fof(t1_boole,axiom,(
! [A] : set_union2(A,empty_set) = A )).

fof(t1_xboole_1,lemma,(
! [A,B,C] :
( ( subset(A,B)
& subset(B,C) )
=> subset(A,C) ) )).

fof(t26_xboole_1,lemma,(
! [A,B,C] :
( subset(A,B)
=> subset(set_intersection2(A,C),set_intersection2(B,C)) ) )).

fof(t28_xboole_1,lemma,(
! [A,B] :
( subset(A,B)
=> set_intersection2(A,B) = A ) )).

fof(t2_boole,axiom,(
! [A] : set_intersection2(A,empty_set) = empty_set )).

fof(t2_tarski,axiom,(
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ) )).

fof(t2_xboole_1,lemma,(
! [A] : subset(empty_set,A) )).

fof(t33_xboole_1,lemma,(
! [A,B,C] :
( subset(A,B)
=> subset(set_difference(A,C),set_difference(B,C)) ) )).

fof(t36_xboole_1,lemma,(
! [A,B] : subset(set_difference(A,B),A) )).

fof(t37_xboole_1,lemma,(
! [A,B] :
( set_difference(A,B) = empty_set
<=> subset(A,B) ) )).

fof(t39_xboole_1,lemma,(
! [A,B] : set_union2(A,set_difference(B,A)) = set_union2(A,B) )).

fof(t3_boole,axiom,(
! [A] : set_difference(A,empty_set) = A )).

fof(t3_xboole_0,lemma,(
! [A,B] :
( ~ ( ~ disjoint(A,B)
& ! [C] : ~ ( in(C,A)
& in(C,B) ) )
& ~ ( ? [C] :
( in(C,A)
& in(C,B) )
& disjoint(A,B) ) ) )).

fof(t3_xboole_1,lemma,(
! [A] :
( subset(A,empty_set)
=> A = empty_set ) )).

fof(t40_xboole_1,lemma,(
! [A,B] : set_difference(set_union2(A,B),B) = set_difference(A,B) )).

fof(t45_xboole_1,lemma,(
! [A,B] :
( subset(A,B)
=> B = set_union2(A,set_difference(B,A)) ) )).

fof(t48_xboole_1,lemma,(
! [A,B] : set_difference(A,set_difference(A,B)) = set_intersection2(A,B) )).

fof(t4_boole,axiom,(
! [A] : set_difference(empty_set,A) = empty_set )).

fof(t4_xboole_0,lemma,(
! [A,B] :
( ~ ( ~ disjoint(A,B)
& ! [C] : ~ in(C,set_intersection2(A,B)) )
& ~ ( ? [C] : in(C,set_intersection2(A,B))
& disjoint(A,B) ) ) )).

fof(t60_xboole_1,lemma,(
! [A,B] : ~ ( subset(A,B)
& proper_subset(B,A) ) )).

fof(t63_xboole_1,conjecture,(
! [A,B,C] :
( ( subset(A,B)
& disjoint(B,C) )
=> disjoint(A,C) ) )).

fof(t6_boole,axiom,(
! [A] :
( empty(A)
=> A = empty_set ) )).

fof(t7_boole,axiom,(
! [A,B] : ~ ( in(A,B)
& empty(B) ) )).

fof(t7_xboole_1,lemma,(
! [A,B] : subset(A,set_union2(A,B)) )).

fof(t8_boole,axiom,(
! [A,B] : ~ ( empty(A)
& A != B
& empty(B) ) )).

fof(t8_xboole_1,lemma,(
! [A,B,C] :
( ( subset(A,B)
& subset(C,B) )
=> subset(set_union2(A,C),B) ) )).

%------------------------------------------------------------------------------
```