TPTP Problem File: SEU301+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU301+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem s1_ordinal2__e18_27__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-s1_ordinal2__e18_27__finset_1 [Urb07]
% Status : Theorem
% Rating : 0.81 v8.2.0, 0.78 v7.4.0, 0.73 v7.3.0, 0.69 v7.1.0, 0.65 v7.0.0, 0.67 v6.4.0, 0.69 v6.3.0, 0.71 v6.2.0, 0.84 v6.1.0, 0.90 v6.0.0, 0.83 v5.5.0, 0.85 v5.4.0, 0.86 v5.3.0, 0.89 v5.2.0, 0.80 v5.1.0, 0.81 v5.0.0, 0.83 v3.7.0, 0.80 v3.5.0, 0.84 v3.3.0
% Syntax : Number of formulae : 65 ( 20 unt; 0 def)
% Number of atoms : 250 ( 27 equ)
% Maximal formula atoms : 49 ( 3 avg)
% Number of connectives : 239 ( 54 ~; 2 |; 117 &)
% ( 0 <=>; 66 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 2 con; 0-2 aty)
% Number of variables : 104 ( 83 !; 21 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_ordinal2__e18_27__finset_1,conjecture,
( ( ( 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(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(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(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( relation(A)
& empty(A)
& function(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(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(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(rc2_relat_1,axiom,
? [A] :
( ~ empty(A)
& relation(A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(cc2_arytm_3,axiom,
! [A] :
( ( empty(A)
& ordinal(A) )
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ) ).
fof(rc1_arytm_3,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ).
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(cc2_ordinal1,axiom,
! [A] :
( ( epsilon_transitive(A)
& epsilon_connected(A) )
=> ordinal(A) ) ).
fof(rc1_ordinal1,axiom,
? [A] :
( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(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(cc3_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(rc3_ordinal1,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(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(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(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(dt_k1_ordinal1,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k5_ordinal2,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(fc1_ordinal2,axiom,
( epsilon_transitive(omega)
& epsilon_connected(omega)
& ordinal(omega)
& ~ empty(omega) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(cc1_arytm_3,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( element(B,A)
=> ( epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B) ) ) ) ).
fof(cc3_arytm_3,axiom,
! [A] :
( element(A,omega)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A)
& natural(A) ) ) ).
fof(fc1_ordinal1,axiom,
! [A] : ~ empty(succ(A)) ).
fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(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(fc3_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( ~ empty(succ(A))
& epsilon_transitive(succ(A))
& epsilon_connected(succ(A))
& ordinal(succ(A)) ) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(s2_ordinal1__e18_27__finset_1__1,axiom,
( ! [A] :
( ordinal(A)
=> ( ! [B] :
( ordinal(B)
=> ( in(B,A)
=> ( in(B,omega)
=> ! [C] :
( element(C,powerset(powerset(B)))
=> ~ ( C != empty_set
& ! [D] :
~ ( in(D,C)
& ! [E] :
( ( in(E,C)
& subset(D,E) )
=> E = D ) ) ) ) ) ) )
=> ( in(A,omega)
=> ! [F] :
( element(F,powerset(powerset(A)))
=> ~ ( F != empty_set
& ! [G] :
~ ( in(G,F)
& ! [H] :
( ( in(H,F)
& subset(G,H) )
=> H = G ) ) ) ) ) ) )
=> ! [A] :
( ordinal(A)
=> ( in(A,omega)
=> ! [I] :
( element(I,powerset(powerset(A)))
=> ~ ( I != empty_set
& ! [J] :
~ ( in(J,I)
& ! [K] :
( ( in(K,I)
& subset(J,K) )
=> K = J ) ) ) ) ) ) ) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(d1_ordinal1,axiom,
! [A] : succ(A) = set_union2(A,singleton(A)) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
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_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(t10_ordinal1,axiom,
! [A] : in(A,succ(A)) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t42_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( ~ ( ~ being_limit_ordinal(A)
& ! [B] :
( ordinal(B)
=> A != succ(B) ) )
& ~ ( ? [B] :
( ordinal(B)
& A = succ(B) )
& being_limit_ordinal(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) ) ).
%------------------------------------------------------------------------------