TPTP Problem File: NUM413+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM413+1 : TPTP v8.2.0. Released v3.2.0.
% Domain : Number Theory (Ordinals)
% Problem : Ordinal numbers, theorem 49
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [Ban90] Bancerek (1990), The Ordinal Numbers
% [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : ordinal1__t49_ordinal1 [Urb06]
% Status : Theorem
% Rating : 1.00 v7.0.0, 0.93 v6.4.0, 1.00 v5.0.0, 0.96 v4.0.1, 0.91 v4.0.0, 0.92 v3.7.0, 0.95 v3.3.0, 0.93 v3.2.0
% Syntax : Number of formulae : 69 ( 9 unt; 0 def)
% Number of atoms : 245 ( 19 equ)
% Maximal formula atoms : 23 ( 3 avg)
% Number of connectives : 193 ( 17 ~; 4 |; 97 &)
% ( 18 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-2 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-2 aty)
% Number of variables : 130 ( 109 !; 21 ?)
% 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_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_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_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(connectedness_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
| ordinal_subset(B,A) ) ) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( subset(A,B)
& subset(B,A) ) ) ).
fof(d1_funct_1,axiom,
! [A] :
( function(A)
<=> ! [B,C,D] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(B,D),A) )
=> C = D ) ) ).
fof(d1_relat_1,axiom,
! [A] :
( relation(A)
<=> ! [B] :
~ ( in(B,A)
& ! [C,D] : B != ordered_pair(C,D) ) ) ).
fof(d2_ordinal1,axiom,
! [A] :
( epsilon_transitive(A)
<=> ! [B] :
( in(B,A)
=> subset(B,A) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) ).
fof(d4_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( B = relation_dom(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] : in(ordered_pair(C,D),A) ) ) ) ).
fof(d4_tarski,axiom,
! [A,B] :
( B = union(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(C,D)
& in(D,A) ) ) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(D,relation_dom(A))
& C = apply(A,D) ) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(d7_ordinal1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( transfinite_sequence(A)
<=> ordinal(relation_dom(A)) ) ) ).
fof(d9_ordinal1,axiom,
! [A] :
( inclusion_linear(A)
<=> ! [B,C] :
( ( in(B,A)
& in(C,A) )
=> inclusion_comparable(B,C) ) ) ).
fof(d9_xboole_0,axiom,
! [A,B] :
( inclusion_comparable(A,B)
<=> ( subset(A,B)
| subset(B,A) ) ) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
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(fc4_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(union(A))
& epsilon_connected(union(A))
& ordinal(union(A)) ) ) ).
fof(fc4_relat_1,axiom,
( empty(empty_set)
& relation(empty_set) ) ).
fof(fc5_ordinal1,axiom,
! [A] :
( ( relation(A)
& function(A)
& transfinite_sequence(A) )
=> ( epsilon_transitive(relation_dom(A))
& epsilon_connected(relation_dom(A))
& ordinal(relation_dom(A)) ) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc6_funct_1,axiom,
! [A] :
( ( relation(A)
& relation_non_empty(A)
& function(A) )
=> with_non_empty_elements(relation_rng(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(rc1_funct_1,axiom,
? [A] :
( relation(A)
& function(A) ) ).
fof(rc1_ordinal1,axiom,
? [A] :
( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc1_relat_1,axiom,
? [A] :
( empty(A)
& relation(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
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_xboole_0,axiom,
? [A] : ~ empty(A) ).
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(rc4_ordinal1,axiom,
? [A] :
( relation(A)
& function(A)
& transfinite_sequence(A) ) ).
fof(rc5_funct_1,axiom,
? [A] :
( relation(A)
& relation_non_empty(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(reflexivity_r3_xboole_0,axiom,
! [A,B] : inclusion_comparable(A,A) ).
fof(s1_funct_1__e6_60__ordinal1,axiom,
! [A] :
( ! [B,C,D] :
( ( in(B,A)
& ! [E] :
( ( relation(E)
& function(E)
& transfinite_sequence(E) )
=> ( E = B
=> relation_dom(E) = C ) )
& in(B,A)
& ! [F] :
( ( relation(F)
& function(F)
& transfinite_sequence(F) )
=> ( F = B
=> relation_dom(F) = D ) ) )
=> C = D )
=> ? [B] :
( relation(B)
& function(B)
& ! [C,D] :
( in(ordered_pair(C,D),B)
<=> ( in(C,A)
& in(C,A)
& ! [G] :
( ( relation(G)
& function(G)
& transfinite_sequence(G) )
=> ( G = C
=> relation_dom(G) = D ) ) ) ) ) ) ).
fof(s1_ordinal1__e14_60__ordinal1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( ? [B] :
( ordinal(B)
& ! [C] :
( ordinal(C)
=> ( in(C,relation_rng(A))
=> ordinal_subset(C,B) ) ) )
=> ? [B] :
( ordinal(B)
& ! [D] :
( ordinal(D)
=> ( in(D,relation_rng(A))
=> ordinal_subset(D,B) ) )
& ! [E] :
( ordinal(E)
=> ( ! [F] :
( ordinal(F)
=> ( in(F,relation_rng(A))
=> ordinal_subset(F,E) ) )
=> ordinal_subset(B,E) ) ) ) ) ) ).
fof(symmetry_r3_xboole_0,axiom,
! [A,B] :
( inclusion_comparable(A,B)
=> inclusion_comparable(B,A) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t23_ordinal1,axiom,
! [A,B] :
( ordinal(B)
=> ( in(A,B)
=> ordinal(A) ) ) ).
fof(t26_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ! [B] :
( ordinal(B)
=> ( ordinal_subset(A,B)
| in(B,A) ) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t36_ordinal1,axiom,
! [A] :
~ ( ! [B] :
( in(B,A)
=> ordinal(B) )
& ! [B] :
( ordinal(B)
=> ~ subset(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t49_ordinal1,conjecture,
! [A] :
( ( ! [B] :
( in(B,A)
=> ( relation(B)
& function(B)
& transfinite_sequence(B) ) )
& inclusion_linear(A) )
=> ( relation(union(A))
& function(union(A))
& transfinite_sequence(union(A)) ) ) ).
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) ) ).
fof(t8_funct_1,axiom,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(ordered_pair(A,B),C)
<=> ( in(A,relation_dom(C))
& B = apply(C,A) ) ) ) ).
fof(t92_zfmisc_1,axiom,
! [A,B] :
( in(A,B)
=> subset(A,union(B)) ) ).
%------------------------------------------------------------------------------