TPTP Problem File: SEU302+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU302+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t18_finset_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-t18_finset_1 [Urb07]
% Status : Theorem
% Rating : 1.00 v3.7.0, 0.95 v3.5.0, 1.00 v3.3.0
% Syntax : Number of formulae : 112 ( 30 unt; 0 def)
% Number of atoms : 361 ( 40 equ)
% Maximal formula atoms : 49 ( 3 avg)
% Number of connectives : 305 ( 56 ~; 5 |; 141 &)
% ( 17 <=>; 86 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 2 con; 0-2 aty)
% Number of variables : 187 ( 163 !; 24 ?)
% 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(cc1_arytm_3,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( element(B,A)
=> ( epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B) ) ) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc2_arytm_3,axiom,
! [A] :
( ( empty(A)
& ordinal(A) )
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(cc2_ordinal1,axiom,
! [A] :
( ( epsilon_transitive(A)
& epsilon_connected(A) )
=> ordinal(A) ) ).
fof(cc3_arytm_3,axiom,
! [A] :
( element(A,omega)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ) ).
fof(cc3_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(connectedness_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
| ordinal_subset(B,A) ) ) ).
fof(d1_finset_1,axiom,
! [A] :
( finite(A)
<=> ? [B] :
( relation(B)
& function(B)
& relation_rng(B) = A
& in(relation_dom(B),omega) ) ) ).
fof(d1_ordinal1,axiom,
! [A] : succ(A) = set_union2(A,singleton(A)) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ) ).
fof(d2_ordinal1,axiom,
! [A] :
( epsilon_transitive(A)
<=> ! [B] :
( in(B,A)
=> subset(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(d4_xboole_0,axiom,
! [A,B,C] :
( C = set_difference(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
& ~ in(D,B) ) ) ) ).
fof(d5_ordinal2,axiom,
! [A] :
( A = omega
<=> ( in(empty_set,A)
& being_limit_ordinal(A)
& ordinal(A)
& ! [B] :
( ordinal(B)
=> ( ( in(empty_set,B)
& being_limit_ordinal(B) )
=> subset(A,B) ) ) ) ) ).
fof(dt_k10_relat_1,axiom,
$true ).
fof(dt_k1_ordinal1,axiom,
$true ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_relat_1,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k4_xboole_0,axiom,
$true ).
fof(dt_k5_ordinal2,axiom,
$true ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc12_finset_1,axiom,
! [A,B] :
( finite(A)
=> finite(set_difference(A,B)) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc13_finset_1,axiom,
! [A,B] :
( ( relation(A)
& function(A)
& finite(B) )
=> finite(relation_image(A,B)) ) ).
fof(fc1_finset_1,axiom,
! [A] :
( ~ empty(singleton(A))
& finite(singleton(A)) ) ).
fof(fc1_ordinal1,axiom,
! [A] : ~ empty(succ(A)) ).
fof(fc1_ordinal2,axiom,
( epsilon_transitive(omega)
& epsilon_connected(omega)
& ordinal(omega)
& ~ empty(omega) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_arytm_3,axiom,
! [A] :
( ( ordinal(A)
& natural(A) )
=> ( ~ empty(succ(A))
& epsilon_transitive(succ(A))
& epsilon_connected(succ(A))
& ordinal(succ(A))
& natural(succ(A)) ) ) ).
fof(fc2_ordinal1,axiom,
( relation(empty_set)
& relation_empty_yielding(empty_set)
& function(empty_set)
& one_to_one(empty_set)
& empty(empty_set)
& epsilon_transitive(empty_set)
& epsilon_connected(empty_set)
& ordinal(empty_set) ) ).
fof(fc2_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(set_union2(A,B)) ) ).
fof(fc2_subset_1,axiom,
! [A] : ~ empty(singleton(A)) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( ~ empty(succ(A))
& epsilon_transitive(succ(A))
& epsilon_connected(succ(A))
& ordinal(succ(A)) ) ) ).
fof(fc3_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(set_difference(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_dom(A))
& relation(relation_dom(A)) ) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_rng(A))
& relation(relation_rng(A)) ) ) ).
fof(fc9_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(set_union2(A,B)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(rc1_arytm_3,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_ordinal1,axiom,
? [A] :
( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc1_ordinal2,axiom,
? [A] :
( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& being_limit_ordinal(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(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_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(rc2_ordinal1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A)
& empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
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(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc3_ordinal1,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(A) ) ).
fof(redefinition_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
<=> subset(A,B) ) ) ).
fof(reflexivity_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ordinal_subset(A,A) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(s1_ordinal2__e18_27__finset_1,axiom,
( ( ( in(empty_set,omega)
=> ! [A] :
( element(A,powerset(powerset(empty_set)))
=> ~ ( A != empty_set
& ! [B] :
~ ( in(B,A)
& ! [C] :
( ( in(C,A)
& subset(B,C) )
=> C = B ) ) ) ) )
& ! [D] :
( ordinal(D)
=> ( ( in(D,omega)
=> ! [E] :
( element(E,powerset(powerset(D)))
=> ~ ( E != empty_set
& ! [F] :
~ ( in(F,E)
& ! [G] :
( ( in(G,E)
& subset(F,G) )
=> G = F ) ) ) ) )
=> ( in(succ(D),omega)
=> ! [H] :
( element(H,powerset(powerset(succ(D))))
=> ~ ( H != empty_set
& ! [I] :
~ ( in(I,H)
& ! [J] :
( ( in(J,H)
& subset(I,J) )
=> J = I ) ) ) ) ) ) )
& ! [D] :
( ordinal(D)
=> ( ( being_limit_ordinal(D)
& ! [K] :
( ordinal(K)
=> ( in(K,D)
=> ( in(K,omega)
=> ! [L] :
( element(L,powerset(powerset(K)))
=> ~ ( L != empty_set
& ! [M] :
~ ( in(M,L)
& ! [N] :
( ( in(N,L)
& subset(M,N) )
=> N = M ) ) ) ) ) ) ) )
=> ( D = empty_set
| ( in(D,omega)
=> ! [O] :
( element(O,powerset(powerset(D)))
=> ~ ( O != empty_set
& ! [P] :
~ ( in(P,O)
& ! [Q] :
( ( in(Q,O)
& subset(P,Q) )
=> Q = P ) ) ) ) ) ) ) ) )
=> ! [D] :
( ordinal(D)
=> ( in(D,omega)
=> ! [R] :
( element(R,powerset(powerset(D)))
=> ~ ( R != empty_set
& ! [S] :
~ ( in(S,R)
& ! [T] :
( ( in(T,R)
& subset(S,T) )
=> T = S ) ) ) ) ) ) ) ).
fof(s1_xboole_0__e4_27_3_1__finset_1,axiom,
! [A,B] :
( ( ordinal(A)
& element(B,powerset(powerset(succ(A)))) )
=> ? [C] :
! [D] :
( in(D,C)
<=> ( in(D,powerset(A))
& ? [E] :
( in(E,B)
& D = set_difference(E,singleton(A)) ) ) ) ) ).
fof(s1_xboole_0__e6_27__finset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(powerset(A)))
& relation(C)
& function(C) )
=> ? [D] :
! [E] :
( in(E,D)
<=> ( in(E,powerset(relation_dom(C)))
& in(relation_image(C,E),B) ) ) ) ).
fof(t10_ordinal1,axiom,
! [A] : in(A,succ(A)) ).
fof(t146_funct_1,axiom,
! [A,B] :
( relation(B)
=> ( subset(A,relation_dom(B))
=> subset(A,relation_inverse_image(B,relation_image(B,A))) ) ) ).
fof(t147_funct_1,axiom,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( subset(A,relation_rng(B))
=> relation_image(B,relation_inverse_image(B,A)) = A ) ) ).
fof(t167_relat_1,axiom,
! [A,B] :
( relation(B)
=> subset(relation_inverse_image(B,A),relation_dom(B)) ) ).
fof(t178_relat_1,axiom,
! [A,B,C] :
( relation(C)
=> ( subset(A,B)
=> subset(relation_inverse_image(C,A),relation_inverse_image(C,B)) ) ) ).
fof(t18_finset_1,conjecture,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(powerset(A)))
=> ~ ( B != empty_set
& ! [C] :
~ ( in(C,B)
& ! [D] :
( ( in(D,B)
& subset(C,D) )
=> D = C ) ) ) ) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t1_xboole_1,axiom,
! [A,B,C] :
( ( subset(A,B)
& subset(B,C) )
=> subset(A,C) ) ).
fof(t1_zfmisc_1,axiom,
powerset(empty_set) = singleton(empty_set) ).
fof(t23_ordinal1,axiom,
! [A,B] :
( ordinal(B)
=> ( in(A,B)
=> ordinal(A) ) ) ).
fof(t24_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( ordinal(B)
=> ~ ( ~ in(A,B)
& A != B
& ~ in(B,A) ) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t33_xboole_1,axiom,
! [A,B,C] :
( subset(A,B)
=> subset(set_difference(A,C),set_difference(B,C)) ) ).
fof(t37_zfmisc_1,axiom,
! [A,B] :
( subset(singleton(A),B)
<=> in(A,B) ) ).
fof(t39_xboole_1,axiom,
! [A,B] : set_union2(A,set_difference(B,A)) = set_union2(A,B) ).
fof(t39_zfmisc_1,axiom,
! [A,B] :
( subset(A,singleton(B))
<=> ( A = empty_set
| A = singleton(B) ) ) ).
fof(t3_boole,axiom,
! [A] : set_difference(A,empty_set) = A ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t40_xboole_1,axiom,
! [A,B] : set_difference(set_union2(A,B),B) = set_difference(A,B) ).
fof(t45_xboole_1,axiom,
! [A,B] :
( subset(A,B)
=> B = set_union2(A,set_difference(B,A)) ) ).
fof(t46_zfmisc_1,axiom,
! [A,B] :
( in(A,B)
=> set_union2(singleton(A),B) = B ) ).
fof(t4_boole,axiom,
! [A] : set_difference(empty_set,A) = empty_set ).
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(t65_zfmisc_1,axiom,
! [A,B] :
( set_difference(A,singleton(B)) = A
<=> ~ in(B,A) ) ).
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) ) ).
%------------------------------------------------------------------------------