TPTP Problem File: SEU111+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU111+1 : TPTP v8.2.0. Released v3.2.0.
% Domain : Set theory
% Problem : Boolean domains, theorem 24
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [TD90] Trybulec & Darmochwal (1990), Boolean Domains
% : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : finsub_1__t24_finsub_1 [Urb06]
% Status : Theorem
% Rating : 0.83 v8.2.0, 0.86 v7.5.0, 0.84 v7.4.0, 0.83 v7.3.0, 0.90 v7.2.0, 0.86 v7.1.0, 0.91 v7.0.0, 0.83 v6.4.0, 0.81 v6.3.0, 0.88 v6.2.0, 0.92 v6.1.0, 1.00 v6.0.0, 0.96 v5.2.0, 0.95 v5.1.0, 0.90 v5.0.0, 0.88 v4.1.0, 0.83 v4.0.1, 0.87 v4.0.0, 0.83 v3.7.0, 0.75 v3.5.0, 0.84 v3.4.0, 0.89 v3.3.0, 0.86 v3.2.0
% Syntax : Number of formulae : 43 ( 12 unt; 0 def)
% Number of atoms : 109 ( 9 equ)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 83 ( 17 ~; 1 |; 38 &)
% ( 7 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-2 aty)
% Number of functors : 4 ( 4 usr; 1 con; 0-2 aty)
% Number of variables : 78 ( 68 !; 10 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
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fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(cc1_finsub_1,axiom,
! [A] :
( preboolean(A)
=> ( cup_closed(A)
& diff_closed(A) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc2_finsub_1,axiom,
! [A] :
( ( cup_closed(A)
& diff_closed(A) )
=> preboolean(A) ) ).
fof(cc3_finsub_1,axiom,
! [A,B] :
( element(B,finite_subsets(A))
=> finite(B) ) ).
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(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(d5_finsub_1,axiom,
! [A,B] :
( preboolean(B)
=> ( B = finite_subsets(A)
<=> ! [C] :
( in(C,B)
<=> ( subset(C,A)
& finite(C) ) ) ) ) ).
fof(dt_k5_finsub_1,axiom,
! [A] : preboolean(finite_subsets(A)) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc10_finset_1,axiom,
! [A,B] :
( finite(B)
=> finite(set_intersection2(A,B)) ) ).
fof(fc11_finset_1,axiom,
! [A,B] :
( finite(A)
=> finite(set_intersection2(A,B)) ) ).
fof(fc1_finsub_1,axiom,
! [A] :
( ~ empty(powerset(A))
& cup_closed(powerset(A))
& diff_closed(powerset(A))
& preboolean(powerset(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_finsub_1,axiom,
! [A] :
( ~ empty(finite_subsets(A))
& cup_closed(finite_subsets(A))
& diff_closed(finite_subsets(A))
& preboolean(finite_subsets(A)) ) ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_finsub_1,axiom,
? [A] :
( ~ empty(A)
& cup_closed(A)
& cap_closed(A)
& diff_closed(A)
& preboolean(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_finset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B)
& relation(B)
& function(B)
& one_to_one(B)
& epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B)
& natural(B)
& finite(B) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t17_xboole_1,axiom,
! [A,B] : subset(set_intersection2(A,B),A) ).
fof(t19_xboole_1,axiom,
! [A,B,C] :
( ( subset(A,B)
& subset(A,C) )
=> subset(A,set_intersection2(B,C)) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t23_finsub_1,axiom,
! [A,B] :
( subset(A,B)
=> subset(finite_subsets(A),finite_subsets(B)) ) ).
fof(t24_finsub_1,conjecture,
! [A,B] : finite_subsets(set_intersection2(A,B)) = set_intersection2(finite_subsets(A),finite_subsets(B)) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
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