TPTP Problem File: SEU310+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU310+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_xboole_0__e1_61_1__subset_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 : bushy-s1_xboole_0__e1_61_1__subset_1 [Urb07]
% Status : Theorem
% Rating : 0.36 v8.2.0, 0.33 v8.1.0, 0.31 v7.5.0, 0.34 v7.4.0, 0.20 v7.3.0, 0.24 v7.2.0, 0.21 v7.1.0, 0.26 v7.0.0, 0.20 v6.4.0, 0.19 v6.3.0, 0.25 v6.2.0, 0.32 v6.1.0, 0.43 v6.0.0, 0.39 v5.5.0, 0.44 v5.4.0, 0.46 v5.3.0, 0.44 v5.2.0, 0.25 v5.1.0, 0.24 v5.0.0, 0.29 v4.1.0, 0.30 v4.0.0, 0.33 v3.7.0, 0.30 v3.5.0, 0.32 v3.4.0, 0.37 v3.3.0
% Syntax : Number of formulae : 36 ( 6 unt; 0 def)
% Number of atoms : 132 ( 4 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 101 ( 5 ~; 0 |; 54 &)
% ( 2 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-2 aty)
% Number of functors : 4 ( 4 usr; 0 con; 1-2 aty)
% Number of variables : 60 ( 53 !; 7 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_xboole_0__e2_37_1_1__pre_topc__1,conjecture,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ? [C] :
! [D] :
( in(D,C)
<=> ( in(D,powerset(the_carrier(A)))
& in(set_difference(cast_as_carrier_subset(A),D),B) ) ) ) ).
fof(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
fof(cc2_membered,axiom,
! [A] :
( v4_membered(A)
=> v3_membered(A) ) ).
fof(cc3_membered,axiom,
! [A] :
( v3_membered(A)
=> v2_membered(A) ) ).
fof(cc4_membered,axiom,
! [A] :
( v2_membered(A)
=> v1_membered(A) ) ).
fof(rc1_membered,axiom,
? [A] :
( ~ empty(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(cc10_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( element(B,A)
=> v1_xcmplx_0(B) ) ) ).
fof(cc11_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B) ) ) ) ).
fof(cc12_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_rat_1(B) ) ) ) ).
fof(cc13_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc14_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( element(B,A)
=> ( v1_xcmplx_0(B)
& natural(B)
& v1_xreal_0(B)
& v1_int_1(B)
& v1_rat_1(B) ) ) ) ).
fof(cc16_membered,axiom,
! [A] :
( v1_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> v1_membered(B) ) ) ).
fof(cc17_membered,axiom,
! [A] :
( v2_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B) ) ) ) ).
fof(cc18_membered,axiom,
! [A] :
( v3_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B) ) ) ) ).
fof(cc19_membered,axiom,
! [A] :
( v4_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B) ) ) ) ).
fof(cc20_membered,axiom,
! [A] :
( v5_membered(A)
=> ! [B] :
( element(B,powerset(A))
=> ( v1_membered(B)
& v2_membered(B)
& v3_membered(B)
& v4_membered(B)
& v5_membered(B) ) ) ) ).
fof(fc37_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(set_difference(A,B)) ) ).
fof(fc38_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(set_difference(A,B))
& v2_membered(set_difference(A,B)) ) ) ).
fof(fc39_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(set_difference(A,B))
& v2_membered(set_difference(A,B))
& v3_membered(set_difference(A,B)) ) ) ).
fof(fc40_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(set_difference(A,B))
& v2_membered(set_difference(A,B))
& v3_membered(set_difference(A,B))
& v4_membered(set_difference(A,B)) ) ) ).
fof(fc41_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(set_difference(A,B))
& v2_membered(set_difference(A,B))
& v3_membered(set_difference(A,B))
& v4_membered(set_difference(A,B))
& v5_membered(set_difference(A,B)) ) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& closed_subset(B,A) ) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(cc15_membered,axiom,
! [A] :
( empty(A)
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> element(cast_as_carrier_subset(A),powerset(the_carrier(A))) ) ).
fof(dt_k4_xboole_0,axiom,
$true ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( top_str(A)
=> one_sorted_str(A) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(fc5_pre_topc,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> closed_subset(cast_as_carrier_subset(A),A) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(s1_tarski__e2_37_1_1__pre_topc__1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ! [C,D,E] :
( ( C = D
& in(set_difference(cast_as_carrier_subset(A),D),B)
& C = E
& in(set_difference(cast_as_carrier_subset(A),E),B) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,powerset(the_carrier(A)))
& E = D
& in(set_difference(cast_as_carrier_subset(A),D),B) ) ) ) ) ).
%------------------------------------------------------------------------------