TPTP Problem File: SEU366+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU366+2 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP chainy problem t1_waybel_0
% 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 : chainy-t1_waybel_0 [Urb07]
% Status : Theorem
% Rating : 1.00 v5.0.0, 0.96 v4.1.0, 1.00 v3.3.0
% Syntax : Number of formulae : 704 ( 81 unt; 0 def)
% Number of atoms : 3316 ( 541 equ)
% Maximal formula atoms : 49 ( 4 avg)
% Number of connectives : 2937 ( 325 ~; 13 |;1338 &)
% ( 230 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 100 ( 98 usr; 1 prp; 0-3 aty)
% Number of functors : 82 ( 82 usr; 2 con; 0-6 aty)
% Number of variables : 1910 (1681 !; 229 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(abstractness_v1_orders_2,axiom,
! [A] :
( rel_str(A)
=> ( strict_rel_str(A)
=> A = rel_str_of(the_carrier(A),the_InternalRel(A)) ) ) ).
fof(abstractness_v3_lattices,axiom,
! [A] :
( latt_str(A)
=> ( strict_latt_str(A)
=> A = latt_str_of(the_carrier(A),the_L_join(A),the_L_meet(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(antisymmetry_r2_xboole_0,axiom,
! [A,B] :
( proper_subset(A,B)
=> ~ proper_subset(B,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(cc15_membered,axiom,
! [A] :
( empty(A)
=> ( v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ) ).
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(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_finsub_1,axiom,
! [A] :
( preboolean(A)
=> ( cup_closed(A)
& diff_closed(A) ) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( empty(A)
=> function(A) ) ).
fof(cc1_funct_2,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& v1_partfun1(C,A,B) )
=> ( function(C)
& quasi_total(C,A,B) ) ) ) ).
fof(cc1_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& lattice(A) )
=> ( ~ empty_carrier(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A) ) ) ) ).
fof(cc1_membered,axiom,
! [A] :
( v5_membered(A)
=> v4_membered(A) ) ).
fof(cc1_ordinal1,axiom,
! [A] :
( ordinal(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).
fof(cc1_partfun1,axiom,
! [A] :
( ( relation(A)
& symmetric(A)
& transitive(A) )
=> ( relation(A)
& reflexive(A) ) ) ).
fof(cc1_relat_1,axiom,
! [A] :
( empty(A)
=> relation(A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( element(C,powerset(cartesian_product2(A,B)))
=> relation(C) ) ).
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(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_finsub_1,axiom,
! [A] :
( ( cup_closed(A)
& diff_closed(A) )
=> preboolean(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( relation(A)
& empty(A)
& function(A) )
=> ( relation(A)
& function(A)
& one_to_one(A) ) ) ).
fof(cc2_funct_2,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& quasi_total(C,A,B)
& bijective(C,A,B) )
=> ( function(C)
& one_to_one(C)
& quasi_total(C,A,B)
& onto(C,A,B) ) ) ) ).
fof(cc2_lattices,axiom,
! [A] :
( latt_str(A)
=> ( ( ~ empty_carrier(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A) )
=> ( ~ empty_carrier(A)
& lattice(A) ) ) ) ).
fof(cc2_membered,axiom,
! [A] :
( v4_membered(A)
=> v3_membered(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_funct_2,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& one_to_one(C)
& quasi_total(C,A,B)
& onto(C,A,B) )
=> ( function(C)
& quasi_total(C,A,B)
& bijective(C,A,B) ) ) ) ).
fof(cc3_membered,axiom,
! [A] :
( v3_membered(A)
=> v2_membered(A) ) ).
fof(cc3_ordinal1,axiom,
! [A] :
( empty(A)
=> ( epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ) ).
fof(cc4_funct_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ( ( function(B)
& v1_partfun1(B,A,A)
& reflexive(B)
& quasi_total(B,A,A) )
=> ( function(B)
& one_to_one(B)
& quasi_total(B,A,A)
& onto(B,A,A)
& bijective(B,A,A) ) ) ) ).
fof(cc4_membered,axiom,
! [A] :
( v2_membered(A)
=> v1_membered(A) ) ).
fof(cc5_funct_2,axiom,
! [A,B] :
( ~ empty(B)
=> ! [C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& quasi_total(C,A,B) )
=> ( function(C)
& v1_partfun1(C,A,B)
& quasi_total(C,A,B) ) ) ) ) ).
fof(cc6_funct_2,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ! [C] :
( relation_of2(C,A,B)
=> ( ( function(C)
& quasi_total(C,A,B) )
=> ( function(C)
& ~ empty(C)
& v1_partfun1(C,A,B)
& quasi_total(C,A,B) ) ) ) ) ).
fof(commutativity_k2_struct_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> unordered_pair_as_carrier_subset(A,B,C) = unordered_pair_as_carrier_subset(A,C,B) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(commutativity_k3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& join_commutative(A)
& join_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> join_commut(A,B,C) = join_commut(A,C,B) ) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
fof(commutativity_k4_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> meet_commut(A,B,C) = meet_commut(A,C,B) ) ).
fof(commutativity_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_union2(A,B,C) = subset_union2(A,C,B) ) ).
fof(commutativity_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,C) = subset_intersection2(A,C,B) ) ).
fof(connectedness_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
| ordinal_subset(B,A) ) ) ).
fof(d10_relat_1,axiom,
! [A,B] :
( relation(B)
=> ( B = identity_relation(A)
<=> ! [C,D] :
( in(ordered_pair(C,D),B)
<=> ( in(C,A)
& C = D ) ) ) ) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( subset(A,B)
& subset(B,A) ) ) ).
fof(d10_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( ex_inf_of_relstr_set(A,B)
=> ( C = meet_on_relstr(A,B)
<=> ( relstr_element_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_element_smaller(A,B,D)
=> related(A,D,C) ) ) ) ) ) ) ) ).
fof(d11_grcat_1,axiom,
! [A] :
( one_sorted_str(A)
=> identity_on_carrier(A) = identity_as_relation_of(the_carrier(A)) ) ).
fof(d11_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B,C] :
( relation(C)
=> ( C = relation_dom_restriction(A,B)
<=> ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ( in(D,B)
& in(ordered_pair(D,E),A) ) ) ) ) ) ).
fof(d11_yellow_0,axiom,
! [A] :
( rel_str(A)
=> bottom_of_relstr(A) = join_on_relstr(A,empty_set) ) ).
fof(d12_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( C = relation_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E] :
( in(E,relation_dom(A))
& in(E,B)
& D = apply(A,E) ) ) ) ) ).
fof(d12_relat_1,axiom,
! [A,B] :
( relation(B)
=> ! [C] :
( relation(C)
=> ( C = relation_rng_restriction(A,B)
<=> ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ( in(E,A)
& in(ordered_pair(D,E),B) ) ) ) ) ) ).
fof(d12_relat_2,axiom,
! [A] :
( relation(A)
=> ( antisymmetric(A)
<=> is_antisymmetric_in(A,relation_field(A)) ) ) ).
fof(d13_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( C = relation_inverse_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,relation_dom(A))
& in(apply(A,D),B) ) ) ) ) ).
fof(d13_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ( lower_bounded_semilattstr(A)
<=> ? [B] :
( element(B,the_carrier(A))
& ! [C] :
( element(C,the_carrier(A))
=> ( meet(A,B,C) = B
& meet(A,C,B) = B ) ) ) ) ) ).
fof(d13_pre_topc,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( C = topstr_closure(A,B)
<=> ! [D] :
( in(D,the_carrier(A))
=> ( in(D,C)
<=> ! [E] :
( element(E,powerset(the_carrier(A)))
=> ~ ( open_subset(E,A)
& in(D,E)
& disjoint(B,E) ) ) ) ) ) ) ) ) ).
fof(d13_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B,C] :
( C = relation_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E] :
( in(ordered_pair(E,D),A)
& in(E,B) ) ) ) ) ).
fof(d13_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( rel_str(B)
=> ( subrelstr(B,A)
<=> ( subset(the_carrier(B),the_carrier(A))
& subset(the_InternalRel(B),the_InternalRel(A)) ) ) ) ) ).
fof(d14_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B,C] :
( C = relation_inverse_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E] :
( in(ordered_pair(D,E),A)
& in(E,B) ) ) ) ) ).
fof(d14_relat_2,axiom,
! [A] :
( relation(A)
=> ( connected(A)
<=> is_connected_in(A,relation_field(A)) ) ) ).
fof(d14_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( subrelstr(B,A)
=> ( full_subrelstr(B,A)
<=> the_InternalRel(B) = relation_restriction_as_relation_of(the_InternalRel(A),the_carrier(B)) ) ) ) ).
fof(d16_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( latt_set_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,C)
=> below(A,B,D) ) ) ) ) ) ).
fof(d16_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ( lower_bounded_semilattstr(A)
=> ! [B] :
( element(B,the_carrier(A))
=> ( B = bottom_of_semilattstr(A)
<=> ! [C] :
( element(C,the_carrier(A))
=> ( meet(A,B,C) = B
& meet(A,C,B) = B ) ) ) ) ) ) ).
fof(d16_relat_2,axiom,
! [A] :
( relation(A)
=> ( transitive(A)
<=> is_transitive_in(A,relation_field(A)) ) ) ).
fof(d17_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( latt_element_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,C)
=> below(A,D,B) ) ) ) ) ) ).
fof(d1_binop_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] : apply_binary(A,B,C) = apply(A,ordered_pair(B,C)) ) ).
fof(d1_connsp_2,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( point_neighbourhood(C,A,B)
<=> in(B,interior(A,C)) ) ) ) ) ).
fof(d1_enumset1,axiom,
! [A,B,C,D] :
( D = unordered_triple(A,B,C)
<=> ! [E] :
( in(E,D)
<=> ~ ( E != A
& E != B
& E != C ) ) ) ).
fof(d1_finset_1,axiom,
! [A] :
( finite(A)
<=> ? [B] :
( relation(B)
& function(B)
& relation_rng(B) = A
& in(relation_dom(B),omega) ) ) ).
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_funct_2,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> ( ( ( B = empty_set
=> A = empty_set )
=> ( quasi_total(C,A,B)
<=> A = relation_dom_as_subset(A,B,C) ) )
& ( B = empty_set
=> ( A = empty_set
| ( quasi_total(C,A,B)
<=> C = empty_set ) ) ) ) ) ).
fof(d1_lattice3,axiom,
! [A,B] :
( ( strict_latt_str(B)
& latt_str(B) )
=> ( B = boole_lattice(A)
<=> ( the_carrier(B) = powerset(A)
& ! [C] :
( element(C,powerset(A))
=> ! [D] :
( element(D,powerset(A))
=> ( apply_binary(the_L_join(B),C,D) = subset_union2(A,C,D)
& apply_binary(the_L_meet(B),C,D) = subset_intersection2(A,C,D) ) ) ) ) ) ) ).
fof(d1_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> join(A,B,C) = apply_binary_as_element(the_carrier(A),the_carrier(A),the_carrier(A),the_L_join(A),B,C) ) ) ) ).
fof(d1_mcart_1,axiom,
! [A] :
( ? [B,C] : A = ordered_pair(B,C)
=> ! [B] :
( B = pair_first(A)
<=> ! [C,D] :
( A = ordered_pair(C,D)
=> B = C ) ) ) ).
fof(d1_ordinal1,axiom,
! [A] : succ(A) = set_union2(A,singleton(A)) ).
fof(d1_pre_topc,axiom,
! [A] :
( top_str(A)
=> ( topological_space(A)
<=> ( in(the_carrier(A),the_topology(A))
& ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( subset(B,the_topology(A))
=> in(union_of_subsets(the_carrier(A),B),the_topology(A)) ) )
& ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( ( in(B,the_topology(A))
& in(C,the_topology(A)) )
=> in(subset_intersection2(the_carrier(A),B,C),the_topology(A)) ) ) ) ) ) ) ).
fof(d1_relat_1,axiom,
! [A] :
( relation(A)
<=> ! [B] :
~ ( in(B,A)
& ! [C,D] : B != ordered_pair(C,D) ) ) ).
fof(d1_relat_2,axiom,
! [A] :
( relation(A)
=> ! [B] :
( is_reflexive_in(A,B)
<=> ! [C] :
( in(C,B)
=> in(ordered_pair(C,C),A) ) ) ) ).
fof(d1_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
<=> subset(C,cartesian_product2(A,B)) ) ).
fof(d1_setfam_1,axiom,
! [A,B] :
( ( A != empty_set
=> ( B = set_meet(A)
<=> ! [C] :
( in(C,B)
<=> ! [D] :
( in(D,A)
=> in(C,D) ) ) ) )
& ( A = empty_set
=> ( B = set_meet(A)
<=> B = empty_set ) ) ) ).
fof(d1_struct_0,axiom,
! [A] :
( one_sorted_str(A)
=> ( empty_carrier(A)
<=> empty(the_carrier(A)) ) ) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ) ).
fof(d1_tops_1,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> interior(A,B) = subset_complement(the_carrier(A),topstr_closure(A,subset_complement(the_carrier(A),B))) ) ) ).
fof(d1_tops_2,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( open_subsets(B,A)
<=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( in(C,B)
=> open_subset(C,A) ) ) ) ) ) ).
fof(d1_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( directed_subset(B,A)
<=> ! [C] :
( element(C,the_carrier(A))
=> ! [D] :
( element(D,the_carrier(A))
=> ~ ( in(C,B)
& in(D,B)
& ! [E] :
( element(E,the_carrier(A))
=> ~ ( in(E,B)
& related(A,C,E)
& related(A,D,E) ) ) ) ) ) ) ) ) ).
fof(d1_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B,C] :
( C = fiber(A,B)
<=> ! [D] :
( in(D,C)
<=> ( D != B
& in(ordered_pair(D,B),A) ) ) ) ) ).
fof(d1_wellord2,axiom,
! [A,B] :
( relation(B)
=> ( B = inclusion_relation(A)
<=> ( relation_field(B) = A
& ! [C,D] :
( ( in(C,A)
& in(D,A) )
=> ( in(ordered_pair(C,D),B)
<=> subset(C,D) ) ) ) ) ) ).
fof(d1_xboole_0,axiom,
! [A] :
( A = empty_set
<=> ! [B] : ~ in(B,A) ) ).
fof(d1_zfmisc_1,axiom,
! [A,B] :
( B = powerset(A)
<=> ! [C] :
( in(C,B)
<=> subset(C,A) ) ) ).
fof(d21_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A)
& latt_str(A) )
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( C = join_of_latt_set(A,B)
<=> ( latt_element_smaller(A,C,B)
& ! [D] :
( element(D,the_carrier(A))
=> ( latt_element_smaller(A,D,B)
=> below(A,C,D) ) ) ) ) ) ) ) ).
fof(d22_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ! [B] : meet_of_latt_set(A,B) = join_of_latt_set(A,a_2_2_lattice3(A,B)) ) ).
fof(d2_compts_1,axiom,
! [A] :
( centered(A)
<=> ( A != empty_set
& ! [B] :
~ ( B != empty_set
& subset(B,A)
& finite(B)
& set_meet(B) = empty_set ) ) ) ).
fof(d2_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> poset_of_lattice(A) = rel_str_of(the_carrier(A),k2_lattice3(A)) ) ).
fof(d2_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> meet(A,B,C) = apply_binary_as_element(the_carrier(A),the_carrier(A),the_carrier(A),the_L_meet(A),B,C) ) ) ) ).
fof(d2_mcart_1,axiom,
! [A] :
( ? [B,C] : A = ordered_pair(B,C)
=> ! [B] :
( B = pair_second(A)
<=> ! [C,D] :
( A = ordered_pair(C,D)
=> B = D ) ) ) ).
fof(d2_ordinal1,axiom,
! [A] :
( epsilon_transitive(A)
<=> ! [B] :
( in(B,A)
=> subset(B,A) ) ) ).
fof(d2_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> empty_carrier_subset(A) = empty_set ) ).
fof(d2_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( A = B
<=> ! [C,D] :
( in(ordered_pair(C,D),A)
<=> in(ordered_pair(C,D),B) ) ) ) ) ).
fof(d2_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
=> ( element(B,A)
<=> in(B,A) ) )
& ( empty(A)
=> ( element(B,A)
<=> empty(B) ) ) ) ).
fof(d2_tarski,axiom,
! [A,B,C] :
( C = unordered_pair(A,B)
<=> ! [D] :
( in(D,C)
<=> ( D = A
| D = B ) ) ) ).
fof(d2_tex_2,axiom,
! [A,B] :
( element(B,A)
=> ( proper_element(B,A)
<=> B != union(A) ) ) ).
fof(d2_tops_2,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( closed_subsets(B,A)
<=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( in(C,B)
=> closed_subset(C,A) ) ) ) ) ) ).
fof(d2_wellord1,axiom,
! [A] :
( relation(A)
=> ( well_founded_relation(A)
<=> ! [B] :
~ ( subset(B,relation_field(A))
& B != empty_set
& ! [C] :
~ ( in(C,B)
& disjoint(fiber(A,C),B) ) ) ) ) ).
fof(d2_xboole_0,axiom,
! [A,B,C] :
( C = set_union2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
| in(D,B) ) ) ) ).
fof(d2_zfmisc_1,axiom,
! [A,B,C] :
( C = cartesian_product2(A,B)
<=> ! [D] :
( in(D,C)
<=> ? [E,F] :
( in(E,A)
& in(F,B)
& D = ordered_pair(E,F) ) ) ) ).
fof(d3_compts_1,axiom,
! [A] :
( top_str(A)
=> ( compact_top_space(A)
<=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ~ ( is_a_cover_of_carrier(A,B)
& open_subsets(B,A)
& ! [C] :
( element(C,powerset(powerset(the_carrier(A))))
=> ~ ( subset(C,B)
& is_a_cover_of_carrier(A,C)
& finite(C) ) ) ) ) ) ) ).
fof(d3_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> cast_to_el_of_LattPOSet(A,B) = B ) ) ).
fof(d3_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( below(A,B,C)
<=> join(A,B,C) = C ) ) ) ) ).
fof(d3_ordinal1,axiom,
! [A] :
( epsilon_connected(A)
<=> ! [B,C] :
~ ( in(B,A)
& in(C,A)
& ~ in(B,C)
& B != C
& ~ in(C,B) ) ) ).
fof(d3_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> cast_as_carrier_subset(A) = the_carrier(A) ) ).
fof(d3_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( subset(A,B)
<=> ! [C,D] :
( in(ordered_pair(C,D),A)
=> in(ordered_pair(C,D),B) ) ) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) ).
fof(d3_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( is_well_founded_in(A,B)
<=> ! [C] :
~ ( subset(C,B)
& C != empty_set
& ! [D] :
~ ( in(D,C)
& disjoint(fiber(A,D),C) ) ) ) ) ).
fof(d3_xboole_0,axiom,
! [A,B,C] :
( C = set_intersection2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
& in(D,B) ) ) ) ).
fof(d4_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( ( in(B,relation_dom(A))
=> ( C = apply(A,B)
<=> in(ordered_pair(B,C),A) ) )
& ( ~ in(B,relation_dom(A))
=> ( C = apply(A,B)
<=> C = empty_set ) ) ) ) ).
fof(d4_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(poset_of_lattice(A)))
=> cast_to_el_of_lattice(A,B) = B ) ) ).
fof(d4_ordinal1,axiom,
! [A] :
( ordinal(A)
<=> ( epsilon_transitive(A)
& epsilon_connected(A) ) ) ).
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_relat_2,axiom,
! [A] :
( relation(A)
=> ! [B] :
( is_antisymmetric_in(A,B)
<=> ! [C,D] :
( ( in(C,B)
& in(D,B)
& in(ordered_pair(C,D),A)
& in(ordered_pair(D,C),A) )
=> C = D ) ) ) ).
fof(d4_subset_1,axiom,
! [A] : cast_to_subset(A) = A ).
fof(d4_tarski,axiom,
! [A,B] :
( B = union(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(C,D)
& in(D,A) ) ) ) ).
fof(d4_wellord1,axiom,
! [A] :
( relation(A)
=> ( well_ordering(A)
<=> ( reflexive(A)
& transitive(A)
& antisymmetric(A)
& connected(A)
& well_founded_relation(A) ) ) ) ).
fof(d4_wellord2,axiom,
! [A,B] :
( equipotent(A,B)
<=> ? [C] :
( relation(C)
& function(C)
& one_to_one(C)
& relation_dom(C) = A
& relation_rng(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(d4_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( lower_bounded_relstr(A)
<=> ? [B] :
( element(B,the_carrier(A))
& relstr_element_smaller(A,the_carrier(A),B) ) ) ) ).
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_orders_2,axiom,
! [A] :
( rel_str(A)
=> ( transitive_relstr(A)
<=> is_transitive_in(the_InternalRel(A),the_carrier(A)) ) ) ).
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(d5_pre_topc,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( open_subset(B,A)
<=> in(B,the_topology(A)) ) ) ) ).
fof(d5_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] : in(ordered_pair(D,C),A) ) ) ) ).
fof(d5_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> subset_complement(A,B) = set_difference(A,B) ) ).
fof(d5_tarski,axiom,
! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).
fof(d5_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( well_orders(A,B)
<=> ( is_reflexive_in(A,B)
& is_transitive_in(A,B)
& is_antisymmetric_in(A,B)
& is_connected_in(A,B)
& is_well_founded_in(A,B) ) ) ) ).
fof(d6_orders_2,axiom,
! [A] :
( rel_str(A)
=> ( antisymmetric_relstr(A)
<=> is_antisymmetric_in(the_InternalRel(A),the_carrier(A)) ) ) ).
fof(d6_ordinal1,axiom,
! [A] :
( being_limit_ordinal(A)
<=> A = union(A) ) ).
fof(d6_pre_topc,axiom,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( closed_subset(B,A)
<=> open_subset(subset_difference(the_carrier(A),cast_as_carrier_subset(A),B),A) ) ) ) ).
fof(d6_relat_1,axiom,
! [A] :
( relation(A)
=> relation_field(A) = set_union2(relation_dom(A),relation_rng(A)) ) ).
fof(d6_relat_2,axiom,
! [A] :
( relation(A)
=> ! [B] :
( is_connected_in(A,B)
<=> ! [C,D] :
~ ( in(C,B)
& in(D,B)
& C != D
& ~ in(ordered_pair(C,D),A)
& ~ in(ordered_pair(D,C),A) ) ) ) ).
fof(d6_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B] : relation_restriction(A,B) = set_intersection2(A,cartesian_product2(B,B)) ) ).
fof(d7_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( B = relation_inverse(A)
<=> ! [C,D] :
( in(ordered_pair(C,D),B)
<=> in(ordered_pair(D,C),A) ) ) ) ) ).
fof(d7_wellord1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( relation_isomorphism(A,B,C)
<=> ( relation_dom(C) = relation_field(A)
& relation_rng(C) = relation_field(B)
& one_to_one(C)
& ! [D,E] :
( in(ordered_pair(D,E),A)
<=> ( in(D,relation_field(A))
& in(E,relation_field(A))
& in(ordered_pair(apply(C,D),apply(C,E)),B) ) ) ) ) ) ) ) ).
fof(d7_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
<=> set_intersection2(A,B) = empty_set ) ).
fof(d7_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( ex_sup_of_relstr_set(A,B)
<=> ? [C] :
( element(C,the_carrier(A))
& relstr_set_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_set_smaller(A,B,D)
=> related(A,C,D) ) )
& ! [D] :
( element(D,the_carrier(A))
=> ( ( relstr_set_smaller(A,B,D)
& ! [E] :
( element(E,the_carrier(A))
=> ( relstr_set_smaller(A,B,E)
=> related(A,D,E) ) ) )
=> D = C ) ) ) ) ) ).
fof(d8_filter_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> relation_of_lattice(A) = a_1_0_filter_1(A) ) ).
fof(d8_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
<=> ! [B,C] :
( ( in(B,relation_dom(A))
& in(C,relation_dom(A))
& apply(A,B) = apply(A,C) )
=> B = C ) ) ) ).
fof(d8_lattice3,axiom,
! [A] :
( rel_str(A)
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( relstr_element_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,B)
=> related(A,C,D) ) ) ) ) ) ).
fof(d8_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> ( meet_absorbing(A)
<=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> join(A,meet(A,B,C),C) = C ) ) ) ) ).
fof(d8_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( is_a_cover_of_carrier(A,B)
<=> cast_as_carrier_subset(A) = union_of_subsets(the_carrier(A),B) ) ) ) ).
fof(d8_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( relation(C)
=> ( C = relation_composition(A,B)
<=> ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ? [F] :
( in(ordered_pair(D,F),A)
& in(ordered_pair(F,E),B) ) ) ) ) ) ) ).
fof(d8_relat_2,axiom,
! [A] :
( relation(A)
=> ! [B] :
( is_transitive_in(A,B)
<=> ! [C,D,E] :
( ( in(C,B)
& in(D,B)
& in(E,B)
& in(ordered_pair(C,D),A)
& in(ordered_pair(D,E),A) )
=> in(ordered_pair(C,E),A) ) ) ) ).
fof(d8_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ! [C] :
( element(C,powerset(powerset(A)))
=> ( C = complements_of_subsets(A,B)
<=> ! [D] :
( element(D,powerset(A))
=> ( in(D,C)
<=> in(subset_complement(A,D),B) ) ) ) ) ) ).
fof(d8_xboole_0,axiom,
! [A,B] :
( proper_subset(A,B)
<=> ( subset(A,B)
& A != B ) ) ).
fof(d8_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( ex_inf_of_relstr_set(A,B)
<=> ? [C] :
( element(C,the_carrier(A))
& relstr_element_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_element_smaller(A,B,D)
=> related(A,D,C) ) )
& ! [D] :
( element(D,the_carrier(A))
=> ( ( relstr_element_smaller(A,B,D)
& ! [E] :
( element(E,the_carrier(A))
=> ( relstr_element_smaller(A,B,E)
=> related(A,E,D) ) ) )
=> D = C ) ) ) ) ) ).
fof(d9_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> function_inverse(A) = relation_inverse(A) ) ) ).
fof(d9_lattice3,axiom,
! [A] :
( rel_str(A)
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( relstr_set_smaller(A,B,C)
<=> ! [D] :
( element(D,the_carrier(A))
=> ( in(D,B)
=> related(A,D,C) ) ) ) ) ) ).
fof(d9_orders_2,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( related(A,B,C)
<=> in(ordered_pair(B,C),the_InternalRel(A)) ) ) ) ) ).
fof(d9_relat_2,axiom,
! [A] :
( relation(A)
=> ( reflexive(A)
<=> is_reflexive_in(A,relation_field(A)) ) ) ).
fof(d9_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B,C] :
( element(C,the_carrier(A))
=> ( ex_sup_of_relstr_set(A,B)
=> ( C = join_on_relstr(A,B)
<=> ( relstr_set_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_set_smaller(A,B,D)
=> related(A,C,D) ) ) ) ) ) ) ) ).
fof(dt_g1_orders_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ( strict_rel_str(rel_str_of(A,B))
& rel_str(rel_str_of(A,B)) ) ) ).
fof(dt_g3_lattices,axiom,
! [A,B,C] :
( ( function(B)
& quasi_total(B,cartesian_product2(A,A),A)
& relation_of2(B,cartesian_product2(A,A),A)
& function(C)
& quasi_total(C,cartesian_product2(A,A),A)
& relation_of2(C,cartesian_product2(A,A),A) )
=> ( strict_latt_str(latt_str_of(A,B,C))
& latt_str(latt_str_of(A,B,C)) ) ) ).
fof(dt_k10_filter_1,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A)
& ~ empty_carrier(B)
& lattice(B)
& latt_str(B)
& element(C,the_carrier(A))
& element(D,the_carrier(B)) )
=> element(k10_filter_1(A,B,C,D),the_carrier(k8_filter_1(A,B))) ) ).
fof(dt_k10_relat_1,axiom,
$true ).
fof(dt_k15_lattice3,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> element(join_of_latt_set(A,B),the_carrier(A)) ) ).
fof(dt_k16_lattice3,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& latt_str(A) )
=> element(meet_of_latt_set(A,B),the_carrier(A)) ) ).
fof(dt_k1_binop_1,axiom,
$true ).
fof(dt_k1_domain_1,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty(B)
& element(C,A)
& element(D,B) )
=> element(ordered_pair_as_product_element(A,B,C,D),cartesian_product2(A,B)) ) ).
fof(dt_k1_enumset1,axiom,
$true ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_lattice3,axiom,
! [A] :
( strict_latt_str(boole_lattice(A))
& latt_str(boole_lattice(A)) ) ).
fof(dt_k1_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& join_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(join(A,B,C),the_carrier(A)) ) ).
fof(dt_k1_mcart_1,axiom,
$true ).
fof(dt_k1_ordinal1,axiom,
$true ).
fof(dt_k1_pcomps_1,axiom,
! [A] : element(k1_pcomps_1(A),powerset(powerset(A))) ).
fof(dt_k1_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> element(empty_carrier_subset(A),powerset(the_carrier(A))) ) ).
fof(dt_k1_relat_1,axiom,
$true ).
fof(dt_k1_setfam_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_toler_1,axiom,
! [A,B] :
( relation(A)
=> relation_of2_as_subset(relation_restriction_as_relation_of(A,B),B,B) ) ).
fof(dt_k1_tops_1,axiom,
! [A,B] :
( ( top_str(A)
& element(B,powerset(the_carrier(A))) )
=> element(interior(A,B),powerset(the_carrier(A))) ) ).
fof(dt_k1_wellord1,axiom,
$true ).
fof(dt_k1_wellord2,axiom,
! [A] : relation(inclusion_relation(A)) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_yellow_0,axiom,
! [A,B] :
( rel_str(A)
=> element(join_on_relstr(A,B),the_carrier(A)) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ empty(A)
& ~ empty(B)
& function(D)
& quasi_total(D,cartesian_product2(A,B),C)
& relation_of2(D,cartesian_product2(A,B),C)
& element(E,A)
& element(F,B) )
=> element(apply_binary_as_element(A,B,C,D,E,F),C) ) ).
fof(dt_k2_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( relation(function_inverse(A))
& function(function_inverse(A)) ) ) ).
fof(dt_k2_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ( reflexive(k2_lattice3(A))
& antisymmetric(k2_lattice3(A))
& transitive(k2_lattice3(A))
& v1_partfun1(k2_lattice3(A),the_carrier(A),the_carrier(A))
& relation_of2_as_subset(k2_lattice3(A),the_carrier(A),the_carrier(A)) ) ) ).
fof(dt_k2_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(meet(A,B,C),the_carrier(A)) ) ).
fof(dt_k2_mcart_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_k2_relat_1,axiom,
$true ).
fof(dt_k2_struct_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(unordered_pair_as_carrier_subset(A,B,C),powerset(the_carrier(A))) ) ).
fof(dt_k2_subset_1,axiom,
! [A] : element(cast_to_subset(A),powerset(A)) ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_wellord1,axiom,
! [A,B] :
( relation(A)
=> relation(relation_restriction(A,B)) ) ).
fof(dt_k2_xboole_0,axiom,
$true ).
fof(dt_k2_yellow_0,axiom,
! [A,B] :
( rel_str(A)
=> element(meet_on_relstr(A,B),the_carrier(A)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ( strict_rel_str(poset_of_lattice(A))
& reflexive_relstr(poset_of_lattice(A))
& transitive_relstr(poset_of_lattice(A))
& antisymmetric_relstr(poset_of_lattice(A))
& rel_str(poset_of_lattice(A)) ) ) ).
fof(dt_k3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& join_commutative(A)
& join_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(join_commut(A,B,C),the_carrier(A)) ) ).
fof(dt_k3_relat_1,axiom,
$true ).
fof(dt_k3_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> element(subset_complement(A,B),powerset(A)) ) ).
fof(dt_k3_tarski,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k3_yellow_0,axiom,
! [A] :
( rel_str(A)
=> element(bottom_of_relstr(A),the_carrier(A)) ) ).
fof(dt_k4_lattice3,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A)
& element(B,the_carrier(A)) )
=> element(cast_to_el_of_LattPOSet(A,B),the_carrier(poset_of_lattice(A))) ) ).
fof(dt_k4_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> element(meet_commut(A,B,C),the_carrier(A)) ) ).
fof(dt_k4_relat_1,axiom,
! [A] :
( relation(A)
=> relation(relation_inverse(A)) ) ).
fof(dt_k4_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_dom_as_subset(A,B,C),powerset(A)) ) ).
fof(dt_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> element(subset_union2(A,B,C),powerset(A)) ) ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_k4_xboole_0,axiom,
$true ).
fof(dt_k5_lattice3,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A)
& element(B,the_carrier(poset_of_lattice(A))) )
=> element(cast_to_el_of_lattice(A,B),the_carrier(A)) ) ).
fof(dt_k5_lattices,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_semilatt_str(A) )
=> element(bottom_of_semilattstr(A),the_carrier(A)) ) ).
fof(dt_k5_ordinal2,axiom,
$true ).
fof(dt_k5_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(relation_composition(A,B)) ) ).
fof(dt_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> element(relation_rng_as_subset(A,B,C),powerset(B)) ) ).
fof(dt_k5_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> element(union_of_subsets(A,B),powerset(A)) ) ).
fof(dt_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> element(subset_intersection2(A,B,C),powerset(A)) ) ).
fof(dt_k6_partfun1,axiom,
! [A] :
( v1_partfun1(identity_as_relation_of(A),A,A)
& relation_of2_as_subset(identity_as_relation_of(A),A,A) ) ).
fof(dt_k6_pre_topc,axiom,
! [A,B] :
( ( top_str(A)
& element(B,powerset(the_carrier(A))) )
=> element(topstr_closure(A,B),powerset(the_carrier(A))) ) ).
fof(dt_k6_relat_1,axiom,
! [A] : relation(identity_relation(A)) ).
fof(dt_k6_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> element(meet_of_subsets(A,B),powerset(A)) ) ).
fof(dt_k6_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> element(subset_difference(A,B,C),powerset(A)) ) ).
fof(dt_k7_grcat_1,axiom,
! [A] :
( one_sorted_str(A)
=> ( function(identity_on_carrier(A))
& quasi_total(identity_on_carrier(A),the_carrier(A),the_carrier(A))
& relation_of2_as_subset(identity_on_carrier(A),the_carrier(A),the_carrier(A)) ) ) ).
fof(dt_k7_relat_1,axiom,
! [A,B] :
( relation(A)
=> relation(relation_dom_restriction(A,B)) ) ).
fof(dt_k7_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> element(complements_of_subsets(A,B),powerset(powerset(A))) ) ).
fof(dt_k8_filter_1,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& latt_str(A)
& ~ empty_carrier(B)
& latt_str(B) )
=> ( strict_latt_str(k8_filter_1(A,B))
& latt_str(k8_filter_1(A,B)) ) ) ).
fof(dt_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& function(C)
& quasi_total(C,A,B)
& relation_of2(C,A,B)
& element(D,A) )
=> element(apply_as_element(A,B,C,D),B) ) ).
fof(dt_k8_relat_1,axiom,
! [A,B] :
( relation(B)
=> relation(relation_rng_restriction(A,B)) ) ).
fof(dt_k9_filter_1,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> relation(relation_of_lattice(A)) ) ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_l1_lattices,axiom,
! [A] :
( meet_semilatt_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_pre_topc,axiom,
! [A] :
( top_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_lattices,axiom,
! [A] :
( join_semilatt_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l3_lattices,axiom,
! [A] :
( latt_str(A)
=> ( meet_semilatt_str(A)
& join_semilatt_str(A) ) ) ).
fof(dt_m1_connsp_2,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& element(B,the_carrier(A)) )
=> ! [C] :
( point_neighbourhood(C,A,B)
=> element(C,powerset(the_carrier(A))) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m1_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ! [B] :
( subrelstr(B,A)
=> rel_str(B) ) ) ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> element(C,powerset(cartesian_product2(A,B))) ) ).
fof(dt_u1_lattices,axiom,
! [A] :
( meet_semilatt_str(A)
=> ( function(the_L_meet(A))
& quasi_total(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& relation_of2_as_subset(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(dt_u1_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(dt_u1_pre_topc,axiom,
! [A] :
( top_str(A)
=> element(the_topology(A),powerset(powerset(the_carrier(A)))) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(dt_u2_lattices,axiom,
! [A] :
( join_semilatt_str(A)
=> ( function(the_L_join(A))
& quasi_total(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& relation_of2_as_subset(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(existence_l1_lattices,axiom,
? [A] : meet_semilatt_str(A) ).
fof(existence_l1_orders_2,axiom,
? [A] : rel_str(A) ).
fof(existence_l1_pre_topc,axiom,
? [A] : top_str(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_l2_lattices,axiom,
? [A] : join_semilatt_str(A) ).
fof(existence_l3_lattices,axiom,
? [A] : latt_str(A) ).
fof(existence_m1_connsp_2,axiom,
! [A,B] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A)
& element(B,the_carrier(A)) )
=> ? [C] : point_neighbourhood(C,A,B) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : relation_of2(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(existence_m1_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ? [B] : subrelstr(B,A) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : relation_of2_as_subset(C,A,B) ).
fof(fc10_finset_1,axiom,
! [A,B] :
( finite(B)
=> finite(set_intersection2(A,B)) ) ).
fof(fc10_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(B,A))
& relation(relation_composition(B,A)) ) ) ).
fof(fc11_finset_1,axiom,
! [A,B] :
( finite(A)
=> finite(set_intersection2(A,B)) ) ).
fof(fc11_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_inverse(A))
& relation(relation_inverse(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(fc13_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation_empty_yielding(A) )
=> ( relation(relation_dom_restriction(A,B))
& relation_empty_yielding(relation_dom_restriction(A,B)) ) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(cartesian_product2(A,B)) ) ).
fof(fc1_finset_1,axiom,
! [A] :
( ~ empty(singleton(A))
& finite(singleton(A)) ) ).
fof(fc1_finsub_1,axiom,
! [A] :
( ~ empty(powerset(A))
& cup_closed(powerset(A))
& diff_closed(powerset(A))
& preboolean(powerset(A)) ) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A)
& relation(B)
& function(B) )
=> ( relation(relation_composition(A,B))
& function(relation_composition(A,B)) ) ) ).
fof(fc1_lattice3,axiom,
! [A] :
( ~ empty_carrier(boole_lattice(A))
& strict_latt_str(boole_lattice(A)) ) ).
fof(fc1_orders_2,axiom,
! [A,B] :
( ( ~ empty(A)
& relation_of2(B,A,A) )
=> ( ~ empty_carrier(rel_str_of(A,B))
& strict_rel_str(rel_str_of(A,B)) ) ) ).
fof(fc1_ordinal1,axiom,
! [A] : ~ empty(succ(A)) ).
fof(fc1_ordinal2,axiom,
( epsilon_transitive(omega)
& epsilon_connected(omega)
& ordinal(omega)
& ~ empty(omega) ) ).
fof(fc1_pre_topc,axiom,
! [A] :
( one_sorted_str(A)
=> ( empty(empty_carrier_subset(A))
& v1_membered(empty_carrier_subset(A))
& v2_membered(empty_carrier_subset(A))
& v3_membered(empty_carrier_subset(A))
& v4_membered(empty_carrier_subset(A))
& v5_membered(empty_carrier_subset(A)) ) ) ).
fof(fc1_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(set_intersection2(A,B)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc1_zfmisc_1,axiom,
! [A,B] : ~ empty(ordered_pair(A,B)) ).
fof(fc27_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(set_intersection2(A,B)) ) ).
fof(fc28_membered,axiom,
! [A,B] :
( v1_membered(A)
=> v1_membered(set_intersection2(B,A)) ) ).
fof(fc29_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(set_intersection2(A,B))
& v2_membered(set_intersection2(A,B)) ) ) ).
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_finset_1,axiom,
! [A,B] :
( ~ empty(unordered_pair(A,B))
& finite(unordered_pair(A,B)) ) ).
fof(fc2_funct_1,axiom,
! [A] :
( relation(identity_relation(A))
& function(identity_relation(A)) ) ).
fof(fc2_lattice2,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_commutative(A)
& join_semilatt_str(A) )
=> ( relation(the_L_join(A))
& function(the_L_join(A))
& quasi_total(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& v1_binop_1(the_L_join(A),the_carrier(A))
& v1_partfun1(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(fc2_lattice3,axiom,
! [A] :
( ~ empty_carrier(boole_lattice(A))
& strict_latt_str(boole_lattice(A))
& join_commutative(boole_lattice(A))
& join_associative(boole_lattice(A))
& meet_commutative(boole_lattice(A))
& meet_associative(boole_lattice(A))
& meet_absorbing(boole_lattice(A))
& join_absorbing(boole_lattice(A))
& lattice(boole_lattice(A)) ) ).
fof(fc2_orders_2,axiom,
! [A] :
( ( reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& rel_str(A) )
=> ( relation(the_InternalRel(A))
& reflexive(the_InternalRel(A))
& antisymmetric(the_InternalRel(A))
& transitive(the_InternalRel(A))
& v1_partfun1(the_InternalRel(A),the_carrier(A),the_carrier(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_partfun1,axiom,
! [A] :
( relation(identity_relation(A))
& function(identity_relation(A))
& reflexive(identity_relation(A))
& symmetric(identity_relation(A))
& antisymmetric(identity_relation(A))
& transitive(identity_relation(A)) ) ).
fof(fc2_pre_topc,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(cast_as_carrier_subset(A)) ) ).
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_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> closed_subset(topstr_closure(A,B),A) ) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc30_membered,axiom,
! [A,B] :
( v2_membered(A)
=> ( v1_membered(set_intersection2(B,A))
& v2_membered(set_intersection2(B,A)) ) ) ).
fof(fc31_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(set_intersection2(A,B))
& v2_membered(set_intersection2(A,B))
& v3_membered(set_intersection2(A,B)) ) ) ).
fof(fc32_membered,axiom,
! [A,B] :
( v3_membered(A)
=> ( v1_membered(set_intersection2(B,A))
& v2_membered(set_intersection2(B,A))
& v3_membered(set_intersection2(B,A)) ) ) ).
fof(fc33_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(set_intersection2(A,B))
& v2_membered(set_intersection2(A,B))
& v3_membered(set_intersection2(A,B))
& v4_membered(set_intersection2(A,B)) ) ) ).
fof(fc34_membered,axiom,
! [A,B] :
( v4_membered(A)
=> ( v1_membered(set_intersection2(B,A))
& v2_membered(set_intersection2(B,A))
& v3_membered(set_intersection2(B,A))
& v4_membered(set_intersection2(B,A)) ) ) ).
fof(fc35_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(set_intersection2(A,B))
& v2_membered(set_intersection2(A,B))
& v3_membered(set_intersection2(A,B))
& v4_membered(set_intersection2(A,B))
& v5_membered(set_intersection2(A,B)) ) ) ).
fof(fc36_membered,axiom,
! [A,B] :
( v5_membered(A)
=> ( v1_membered(set_intersection2(B,A))
& v2_membered(set_intersection2(B,A))
& v3_membered(set_intersection2(B,A))
& v4_membered(set_intersection2(B,A))
& v5_membered(set_intersection2(B,A)) ) ) ).
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(fc3_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A)
& one_to_one(A) )
=> ( relation(relation_inverse(A))
& function(relation_inverse(A)) ) ) ).
fof(fc3_lattice2,axiom,
! [A] :
( ( ~ empty_carrier(A)
& join_associative(A)
& join_semilatt_str(A) )
=> ( relation(the_L_join(A))
& function(the_L_join(A))
& quasi_total(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& v2_binop_1(the_L_join(A),the_carrier(A))
& v1_partfun1(the_L_join(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(fc3_lattices,axiom,
! [A,B,C] :
( ( ~ empty(A)
& function(B)
& quasi_total(B,cartesian_product2(A,A),A)
& relation_of2(B,cartesian_product2(A,A),A)
& function(C)
& quasi_total(C,cartesian_product2(A,A),A)
& relation_of2(C,cartesian_product2(A,A),A) )
=> ( ~ empty_carrier(latt_str_of(A,B,C))
& strict_latt_str(latt_str_of(A,B,C)) ) ) ).
fof(fc3_orders_2,axiom,
! [A,B] :
( ( reflexive(B)
& antisymmetric(B)
& transitive(B)
& v1_partfun1(B,A,A)
& relation_of2(B,A,A) )
=> ( strict_rel_str(rel_str_of(A,B))
& reflexive_relstr(rel_str_of(A,B))
& transitive_relstr(rel_str_of(A,B))
& antisymmetric_relstr(rel_str_of(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_subset_1,axiom,
! [A,B] : ~ empty(unordered_pair(A,B)) ).
fof(fc3_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& closed_subset(B,A)
& element(B,powerset(the_carrier(A))) )
=> open_subset(subset_complement(the_carrier(A),B),A) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
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(fc4_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A) )
=> ( relation(relation_dom_restriction(A,B))
& function(relation_dom_restriction(A,B)) ) ) ).
fof(fc4_lattice2,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A) )
=> ( relation(the_L_meet(A))
& function(the_L_meet(A))
& quasi_total(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& v1_binop_1(the_L_meet(A),the_carrier(A))
& v1_partfun1(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(fc4_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ( ~ empty_carrier(poset_of_lattice(A))
& strict_rel_str(poset_of_lattice(A))
& reflexive_relstr(poset_of_lattice(A))
& transitive_relstr(poset_of_lattice(A))
& antisymmetric_relstr(poset_of_lattice(A)) ) ) ).
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(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fc4_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& open_subset(B,A)
& element(B,powerset(the_carrier(A))) )
=> closed_subset(subset_complement(the_carrier(A),B),A) ) ).
fof(fc5_funct_1,axiom,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( relation(relation_rng_restriction(A,B))
& function(relation_rng_restriction(A,B)) ) ) ).
fof(fc5_lattice2,axiom,
! [A] :
( ( ~ empty_carrier(A)
& meet_associative(A)
& meet_semilatt_str(A) )
=> ( relation(the_L_meet(A))
& function(the_L_meet(A))
& quasi_total(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A))
& v2_binop_1(the_L_meet(A),the_carrier(A))
& v1_partfun1(the_L_meet(A),cartesian_product2(the_carrier(A),the_carrier(A)),the_carrier(A)) ) ) ).
fof(fc5_pre_topc,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> closed_subset(cast_as_carrier_subset(A),A) ) ).
fof(fc5_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_dom(A)) ) ).
fof(fc6_membered,axiom,
( empty(empty_set)
& v1_membered(empty_set)
& v2_membered(empty_set)
& v3_membered(empty_set)
& v4_membered(empty_set)
& v5_membered(empty_set) ) ).
fof(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc6_tops_1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> open_subset(interior(A,B),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(fc9_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(A,B))
& relation(relation_composition(A,B)) ) ) ).
fof(fraenkel_a_1_0_filter_1,axiom,
! [A,B] :
( ( ~ empty_carrier(B)
& lattice(B)
& latt_str(B) )
=> ( in(A,a_1_0_filter_1(B))
<=> ? [C,D] :
( element(C,the_carrier(B))
& element(D,the_carrier(B))
& A = ordered_pair_as_product_element(the_carrier(B),the_carrier(B),C,D)
& below_refl(B,C,D) ) ) ) ).
fof(fraenkel_a_2_2_lattice3,axiom,
! [A,B,C] :
( ( ~ empty_carrier(B)
& latt_str(B) )
=> ( in(A,a_2_2_lattice3(B,C))
<=> ? [D] :
( element(D,the_carrier(B))
& A = D
& latt_set_smaller(B,D,C) ) ) ) ).
fof(fraenkel_a_2_3_lattice3,axiom,
! [A,B,C] :
( ( ~ empty_carrier(B)
& lattice(B)
& complete_latt_str(B)
& latt_str(B) )
=> ( in(A,a_2_3_lattice3(B,C))
<=> ? [D] :
( element(D,the_carrier(B))
& A = D
& latt_set_smaller(B,D,C) ) ) ) ).
fof(free_g1_orders_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ! [C,D] :
( rel_str_of(A,B) = rel_str_of(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(free_g3_lattices,axiom,
! [A,B,C] :
( ( function(B)
& quasi_total(B,cartesian_product2(A,A),A)
& relation_of2(B,cartesian_product2(A,A),A)
& function(C)
& quasi_total(C,cartesian_product2(A,A),A)
& relation_of2(C,cartesian_product2(A,A),A) )
=> ! [D,E,F] :
( latt_str_of(A,B,C) = latt_str_of(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(idempotence_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_union2(A,B,B) = B ) ).
fof(idempotence_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,B) = B ) ).
fof(involutiveness_k3_subset_1,axiom,
! [A,B] :
( element(B,powerset(A))
=> subset_complement(A,subset_complement(A,B)) = B ) ).
fof(involutiveness_k4_relat_1,axiom,
! [A] :
( relation(A)
=> relation_inverse(relation_inverse(A)) = A ) ).
fof(involutiveness_k7_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> complements_of_subsets(A,complements_of_subsets(A,B)) = B ) ).
fof(irreflexivity_r2_xboole_0,axiom,
! [A,B] : ~ proper_subset(A,A) ).
fof(l1_wellord1,lemma,
! [A] :
( relation(A)
=> ( reflexive(A)
<=> ! [B] :
( in(B,relation_field(A))
=> in(ordered_pair(B,B),A) ) ) ) ).
fof(l1_zfmisc_1,lemma,
! [A] : singleton(A) != empty_set ).
fof(l23_zfmisc_1,lemma,
! [A,B] :
( in(A,B)
=> set_union2(singleton(A),B) = B ) ).
fof(l25_zfmisc_1,lemma,
! [A,B] :
~ ( disjoint(singleton(A),B)
& in(A,B) ) ).
fof(l28_zfmisc_1,lemma,
! [A,B] :
( ~ in(A,B)
=> disjoint(singleton(A),B) ) ).
fof(l29_wellord1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_dom(relation_rng_restriction(A,B)),relation_dom(B)) ) ).
fof(l2_wellord1,lemma,
! [A] :
( relation(A)
=> ( transitive(A)
<=> ! [B,C,D] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(C,D),A) )
=> in(ordered_pair(B,D),A) ) ) ) ).
fof(l2_zfmisc_1,lemma,
! [A,B] :
( subset(singleton(A),B)
<=> in(A,B) ) ).
fof(l30_wellord2,lemma,
! [A,B] :
( relation(B)
=> ~ ( well_ordering(B)
& equipotent(A,relation_field(B))
& ! [C] :
( relation(C)
=> ~ well_orders(C,A) ) ) ) ).
fof(l32_xboole_1,lemma,
! [A,B] :
( set_difference(A,B) = empty_set
<=> subset(A,B) ) ).
fof(l3_subset_1,lemma,
! [A,B] :
( element(B,powerset(A))
=> ! [C] :
( in(C,B)
=> in(C,A) ) ) ).
fof(l3_wellord1,lemma,
! [A] :
( relation(A)
=> ( antisymmetric(A)
<=> ! [B,C] :
( ( in(ordered_pair(B,C),A)
& in(ordered_pair(C,B),A) )
=> B = C ) ) ) ).
fof(l3_zfmisc_1,lemma,
! [A,B,C] :
( subset(A,B)
=> ( in(C,A)
| subset(A,set_difference(B,singleton(C))) ) ) ).
fof(l40_tops_1,lemma,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( element(C,the_carrier(A))
=> ( in(C,subset_complement(the_carrier(A),B))
<=> ~ in(C,B) ) ) ) ) ).
fof(l4_wellord1,lemma,
! [A] :
( relation(A)
=> ( connected(A)
<=> ! [B,C] :
~ ( in(B,relation_field(A))
& in(C,relation_field(A))
& B != C
& ~ in(ordered_pair(B,C),A)
& ~ in(ordered_pair(C,B),A) ) ) ) ).
fof(l4_zfmisc_1,lemma,
! [A,B] :
( subset(A,singleton(B))
<=> ( A = empty_set
| A = singleton(B) ) ) ).
fof(l50_zfmisc_1,lemma,
! [A,B] :
( in(A,B)
=> subset(A,union(B)) ) ).
fof(l55_zfmisc_1,lemma,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
<=> ( in(A,C)
& in(B,D) ) ) ).
fof(l71_subset_1,lemma,
! [A,B] :
( ! [C] :
( in(C,A)
=> in(C,B) )
=> element(A,powerset(B)) ) ).
fof(l82_funct_1,lemma,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(B,relation_dom(relation_dom_restriction(C,A)))
<=> ( in(B,relation_dom(C))
& in(B,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_funct_2,axiom,
! [A,B] :
? [C] :
( relation_of2(C,A,B)
& relation(C)
& function(C)
& quasi_total(C,A,B) ) ).
fof(rc1_membered,axiom,
? [A] :
( ~ empty(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(rc1_orders_2,axiom,
? [A] :
( rel_str(A)
& strict_rel_str(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_partfun1,axiom,
? [A] :
( relation(A)
& function(A)
& one_to_one(A)
& empty(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_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& open_subset(B,A) ) ) ).
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_funct_2,axiom,
! [A] :
? [B] :
( relation_of2(B,A,A)
& relation(B)
& function(B)
& one_to_one(B)
& quasi_total(B,A,A)
& onto(B,A,A)
& bijective(B,A,A) ) ).
fof(rc2_orders_2,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(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_partfun1,axiom,
! [A,B] :
? [C] :
( relation_of2(C,A,B)
& relation(C)
& function(C) ) ).
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_tex_2,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& ~ proper_element(B,powerset(A)) ) ).
fof(rc2_tops_1,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& open_subset(B,A)
& closed_subset(B,A) ) ) ).
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_lattices,axiom,
? [A] :
( latt_str(A)
& strict_latt_str(A) ) ).
fof(rc3_ordinal1,axiom,
? [A] :
( ~ empty(A)
& epsilon_transitive(A)
& epsilon_connected(A)
& ordinal(A) ) ).
fof(rc3_partfun1,axiom,
! [A] :
? [B] :
( relation_of2(B,A,A)
& relation(B)
& reflexive(B)
& symmetric(B)
& antisymmetric(B)
& transitive(B)
& v1_partfun1(B,A,A) ) ).
fof(rc3_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_funct_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A)
& function(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(rc6_lattices,axiom,
? [A] :
( latt_str(A)
& ~ empty_carrier(A)
& strict_latt_str(A) ) ).
fof(rc6_pre_topc,axiom,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& closed_subset(B,A) ) ) ).
fof(rc7_pre_topc,axiom,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B)
& closed_subset(B,A) ) ) ).
fof(rc9_lattices,axiom,
? [A] :
( latt_str(A)
& ~ empty_carrier(A)
& strict_latt_str(A)
& join_commutative(A)
& join_associative(A)
& meet_commutative(A)
& meet_associative(A)
& meet_absorbing(A)
& join_absorbing(A)
& lattice(A) ) ).
fof(redefinition_k10_filter_1,axiom,
! [A,B,C,D] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A)
& ~ empty_carrier(B)
& lattice(B)
& latt_str(B)
& element(C,the_carrier(A))
& element(D,the_carrier(B)) )
=> k10_filter_1(A,B,C,D) = ordered_pair(C,D) ) ).
fof(redefinition_k1_domain_1,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& ~ empty(B)
& element(C,A)
& element(D,B) )
=> ordered_pair_as_product_element(A,B,C,D) = ordered_pair(C,D) ) ).
fof(redefinition_k1_pcomps_1,axiom,
! [A] : k1_pcomps_1(A) = powerset(A) ).
fof(redefinition_k1_toler_1,axiom,
! [A,B] :
( relation(A)
=> relation_restriction_as_relation_of(A,B) = relation_restriction(A,B) ) ).
fof(redefinition_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ empty(A)
& ~ empty(B)
& function(D)
& quasi_total(D,cartesian_product2(A,B),C)
& relation_of2(D,cartesian_product2(A,B),C)
& element(E,A)
& element(F,B) )
=> apply_binary_as_element(A,B,C,D,E,F) = apply_binary(D,E,F) ) ).
fof(redefinition_k2_lattice3,axiom,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> k2_lattice3(A) = relation_of_lattice(A) ) ).
fof(redefinition_k2_struct_0,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& one_sorted_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> unordered_pair_as_carrier_subset(A,B,C) = unordered_pair(B,C) ) ).
fof(redefinition_k3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& join_commutative(A)
& join_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> join_commut(A,B,C) = join(A,B,C) ) ).
fof(redefinition_k4_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_semilatt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> meet_commut(A,B,C) = meet(A,B,C) ) ).
fof(redefinition_k4_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_dom_as_subset(A,B,C) = relation_dom(C) ) ).
fof(redefinition_k4_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_union2(A,B,C) = set_union2(B,C) ) ).
fof(redefinition_k5_relset_1,axiom,
! [A,B,C] :
( relation_of2(C,A,B)
=> relation_rng_as_subset(A,B,C) = relation_rng(C) ) ).
fof(redefinition_k5_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> union_of_subsets(A,B) = union(B) ) ).
fof(redefinition_k5_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_intersection2(A,B,C) = set_intersection2(B,C) ) ).
fof(redefinition_k6_partfun1,axiom,
! [A] : identity_as_relation_of(A) = identity_relation(A) ).
fof(redefinition_k6_setfam_1,axiom,
! [A,B] :
( element(B,powerset(powerset(A)))
=> meet_of_subsets(A,B) = set_meet(B) ) ).
fof(redefinition_k6_subset_1,axiom,
! [A,B,C] :
( ( element(B,powerset(A))
& element(C,powerset(A)) )
=> subset_difference(A,B,C) = set_difference(B,C) ) ).
fof(redefinition_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ empty(A)
& function(C)
& quasi_total(C,A,B)
& relation_of2(C,A,B)
& element(D,A) )
=> apply_as_element(A,B,C,D) = apply(C,D) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(redefinition_r1_ordinal1,axiom,
! [A,B] :
( ( ordinal(A)
& ordinal(B) )
=> ( ordinal_subset(A,B)
<=> subset(A,B) ) ) ).
fof(redefinition_r2_wellord2,axiom,
! [A,B] :
( equipotent(A,B)
<=> are_equipotent(A,B) ) ).
fof(redefinition_r3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_absorbing(A)
& join_absorbing(A)
& latt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> ( below_refl(A,B,C)
<=> below(A,B,C) ) ) ).
fof(redefinition_r3_orders_2,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& rel_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> ( related_reflexive(A,B,C)
<=> related(A,B,C) ) ) ).
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_r2_wellord2,axiom,
! [A,B] : equipotent(A,A) ).
fof(reflexivity_r3_lattices,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_absorbing(A)
& join_absorbing(A)
& latt_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> below_refl(A,B,B) ) ).
fof(reflexivity_r3_orders_2,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& rel_str(A)
& element(B,the_carrier(A))
& element(C,the_carrier(A)) )
=> related_reflexive(A,B,B) ) ).
fof(s1_funct_1__e10_24__wellord2__1,lemma,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ( ! [C,D,E] :
( ( in(C,A)
& ? [F] :
( C = F
& in(D,F)
& ! [G] :
( in(G,F)
=> in(ordered_pair(D,G),B) ) )
& in(C,A)
& ? [H] :
( C = H
& in(E,H)
& ! [I] :
( in(I,H)
=> in(ordered_pair(E,I),B) ) ) )
=> D = E )
=> ? [C] :
( relation(C)
& function(C)
& ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ( in(D,A)
& in(D,A)
& ? [J] :
( D = J
& in(E,J)
& ! [K] :
( in(K,J)
=> in(ordered_pair(E,K),B) ) ) ) ) ) ) ) ).
fof(s1_funct_1__e16_22__wellord2__1,lemma,
! [A] :
( ! [B,C,D] :
( ( in(B,A)
& C = singleton(B)
& in(B,A)
& D = singleton(B) )
=> C = D )
=> ? [B] :
( relation(B)
& function(B)
& ! [C,D] :
( in(ordered_pair(C,D),B)
<=> ( in(C,A)
& in(C,A)
& D = singleton(C) ) ) ) ) ).
fof(s1_funct_1__e4_7_1__tops_2__1,lemma,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ! [C,D,E] :
( ( in(C,complements_of_subsets(the_carrier(A),B))
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& in(C,complements_of_subsets(the_carrier(A),B))
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
=> ? [C] :
( relation(C)
& function(C)
& ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ( in(D,complements_of_subsets(the_carrier(A),B))
& in(D,complements_of_subsets(the_carrier(A),B))
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = D
=> E = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ) ).
fof(s1_funct_1__e4_7_2__tops_2__1,lemma,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ! [C,D,E] :
( ( in(C,B)
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& in(C,B)
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
=> ? [C] :
( relation(C)
& function(C)
& ! [D,E] :
( in(ordered_pair(D,E),C)
<=> ( in(D,B)
& in(D,B)
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = D
=> E = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ) ).
fof(s1_ordinal1__e8_6__wellord2,lemma,
! [A] :
( ? [B] :
( ordinal(B)
& in(B,A) )
=> ? [B] :
( ordinal(B)
& in(B,A)
& ! [C] :
( ordinal(C)
=> ( in(C,A)
=> ordinal_subset(B,C) ) ) ) ) ).
fof(s1_ordinal2__e18_27__finset_1,lemma,
( ( ( 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_relat_1__e6_21__wellord2,lemma,
! [A,B,C] :
( ( relation(B)
& relation(C)
& function(C) )
=> ? [D] :
( relation(D)
& ! [E,F] :
( in(ordered_pair(E,F),D)
<=> ( in(E,A)
& in(F,A)
& in(ordered_pair(apply(C,E),apply(C,F)),B) ) ) ) ) ).
fof(s1_tarski__e10_24__wellord2__1,axiom,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ( ! [C,D,E] :
( ( in(C,A)
& ? [F] :
( C = F
& in(D,F)
& ! [G] :
( in(G,F)
=> in(ordered_pair(D,G),B) ) )
& in(C,A)
& ? [H] :
( C = H
& in(E,H)
& ! [I] :
( in(I,H)
=> in(ordered_pair(E,I),B) ) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,A)
& in(E,A)
& ? [J] :
( E = J
& in(D,J)
& ! [K] :
( in(K,J)
=> in(ordered_pair(D,K),B) ) ) ) ) ) ) ).
fof(s1_tarski__e10_24__wellord2__2,axiom,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ! [C] :
( ! [D,E,F] :
( ( D = E
& ? [G,H] :
( ordered_pair(G,H) = E
& in(G,A)
& ? [I] :
( G = I
& in(H,I)
& ! [J] :
( in(J,I)
=> in(ordered_pair(H,J),B) ) ) )
& D = F
& ? [K,L] :
( ordered_pair(K,L) = F
& in(K,A)
& ? [M] :
( K = M
& in(L,M)
& ! [N] :
( in(N,M)
=> in(ordered_pair(L,N),B) ) ) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,cartesian_product2(A,C))
& F = E
& ? [O,P] :
( ordered_pair(O,P) = E
& in(O,A)
& ? [Q] :
( O = Q
& in(P,Q)
& ! [R] :
( in(R,Q)
=> in(ordered_pair(P,R),B) ) ) ) ) ) ) ) ).
fof(s1_tarski__e11_2_1__waybel_0__1,axiom,
! [A,B,C] :
( ( ~ empty_carrier(A)
& transitive_relstr(A)
& rel_str(A)
& element(B,powerset(the_carrier(A)))
& finite(C)
& element(C,powerset(B)) )
=> ( ! [D,E,F] :
( ( D = E
& ? [G] :
( G = E
& ? [H] :
( element(H,the_carrier(A))
& in(H,B)
& relstr_set_smaller(A,G,H) ) )
& D = F
& ? [I] :
( I = F
& ? [J] :
( element(J,the_carrier(A))
& in(J,B)
& relstr_set_smaller(A,I,J) ) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,powerset(C))
& F = E
& ? [K] :
( K = E
& ? [L] :
( element(L,the_carrier(A))
& in(L,B)
& relstr_set_smaller(A,K,L) ) ) ) ) ) ) ).
fof(s1_tarski__e16_22__wellord2__1,axiom,
! [A] :
( ! [B,C,D] :
( ( in(B,A)
& C = singleton(B)
& in(B,A)
& D = singleton(B) )
=> C = D )
=> ? [B] :
! [C] :
( in(C,B)
<=> ? [D] :
( in(D,A)
& in(D,A)
& C = singleton(D) ) ) ) ).
fof(s1_tarski__e16_22__wellord2__2,axiom,
! [A,B] :
( ! [C,D,E] :
( ( C = D
& ? [F,G] :
( ordered_pair(F,G) = D
& in(F,A)
& G = singleton(F) )
& C = E
& ? [H,I] :
( ordered_pair(H,I) = E
& in(H,A)
& I = singleton(H) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,cartesian_product2(A,B))
& E = D
& ? [J,K] :
( ordered_pair(J,K) = D
& in(J,A)
& K = singleton(J) ) ) ) ) ).
fof(s1_tarski__e18_27__finset_1__1,axiom,
! [A] :
( ordinal(A)
=> ( ! [B,C,D] :
( ( B = C
& ? [E] :
( ordinal(E)
& C = E
& ( in(E,omega)
=> ! [F] :
( element(F,powerset(powerset(E)))
=> ~ ( F != empty_set
& ! [G] :
~ ( in(G,F)
& ! [H] :
( ( in(H,F)
& subset(G,H) )
=> H = G ) ) ) ) ) )
& B = D
& ? [I] :
( ordinal(I)
& D = I
& ( in(I,omega)
=> ! [J] :
( element(J,powerset(powerset(I)))
=> ~ ( J != empty_set
& ! [K] :
~ ( in(K,J)
& ! [L] :
( ( in(L,J)
& subset(K,L) )
=> L = K ) ) ) ) ) ) )
=> C = D )
=> ? [B] :
! [C] :
( in(C,B)
<=> ? [D] :
( in(D,succ(A))
& D = C
& ? [M] :
( ordinal(M)
& C = M
& ( in(M,omega)
=> ! [N] :
( element(N,powerset(powerset(M)))
=> ~ ( N != empty_set
& ! [O] :
~ ( in(O,N)
& ! [P] :
( ( in(P,N)
& subset(O,P) )
=> P = O ) ) ) ) ) ) ) ) ) ) ).
fof(s1_tarski__e1_40__pre_topc__1,axiom,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> ( ! [C,D,E] :
( ( C = D
& ? [F] :
( element(F,powerset(the_carrier(A)))
& F = D
& closed_subset(F,A)
& subset(B,D) )
& C = E
& ? [G] :
( element(G,powerset(the_carrier(A)))
& G = E
& closed_subset(G,A)
& subset(B,E) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,powerset(the_carrier(A)))
& E = D
& ? [H] :
( element(H,powerset(the_carrier(A)))
& H = D
& closed_subset(H,A)
& subset(B,D) ) ) ) ) ) ).
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) ) ) ) ) ).
fof(s1_tarski__e4_27_3_1__finset_1__1,axiom,
! [A,B] :
( ( ordinal(A)
& element(B,powerset(powerset(succ(A)))) )
=> ( ! [C,D,E] :
( ( C = D
& ? [F] :
( in(F,B)
& D = set_difference(F,singleton(A)) )
& C = E
& ? [G] :
( in(G,B)
& E = set_difference(G,singleton(A)) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,powerset(A))
& E = D
& ? [H] :
( in(H,B)
& D = set_difference(H,singleton(A)) ) ) ) ) ) ).
fof(s1_tarski__e4_7_1__tops_2__1,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ! [C,D,E] :
( ( in(C,complements_of_subsets(the_carrier(A),B))
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& in(C,complements_of_subsets(the_carrier(A),B))
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,complements_of_subsets(the_carrier(A),B))
& in(E,complements_of_subsets(the_carrier(A),B))
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = E
=> D = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ).
fof(s1_tarski__e4_7_1__tops_2__2,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ! [C] :
( ! [D,E,F] :
( ( D = E
& ? [G,H] :
( ordered_pair(G,H) = E
& in(G,complements_of_subsets(the_carrier(A),B))
& ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = G
=> H = subset_complement(the_carrier(A),I) ) ) )
& D = F
& ? [J,K] :
( ordered_pair(J,K) = F
& in(J,complements_of_subsets(the_carrier(A),B))
& ! [L] :
( element(L,powerset(the_carrier(A)))
=> ( L = J
=> K = subset_complement(the_carrier(A),L) ) ) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,cartesian_product2(complements_of_subsets(the_carrier(A),B),C))
& F = E
& ? [M,N] :
( ordered_pair(M,N) = E
& in(M,complements_of_subsets(the_carrier(A),B))
& ! [O] :
( element(O,powerset(the_carrier(A)))
=> ( O = M
=> N = subset_complement(the_carrier(A),O) ) ) ) ) ) ) ) ).
fof(s1_tarski__e4_7_2__tops_2__1,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ! [C,D,E] :
( ( in(C,B)
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& in(C,B)
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,B)
& in(E,B)
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = E
=> D = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ).
fof(s1_tarski__e4_7_2__tops_2__2,axiom,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ! [C] :
( ! [D,E,F] :
( ( D = E
& ? [G,H] :
( ordered_pair(G,H) = E
& in(G,B)
& ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = G
=> H = subset_complement(the_carrier(A),I) ) ) )
& D = F
& ? [J,K] :
( ordered_pair(J,K) = F
& in(J,B)
& ! [L] :
( element(L,powerset(the_carrier(A)))
=> ( L = J
=> K = subset_complement(the_carrier(A),L) ) ) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,cartesian_product2(B,C))
& F = E
& ? [M,N] :
( ordered_pair(M,N) = E
& in(M,B)
& ! [O] :
( element(O,powerset(the_carrier(A)))
=> ( O = M
=> N = subset_complement(the_carrier(A),O) ) ) ) ) ) ) ) ).
fof(s1_tarski__e6_21__wellord2__1,axiom,
! [A,B,C] :
( ( relation(B)
& relation(C)
& function(C) )
=> ( ! [D,E,F] :
( ( D = E
& ? [G,H] :
( E = ordered_pair(G,H)
& in(ordered_pair(apply(C,G),apply(C,H)),B) )
& D = F
& ? [I,J] :
( F = ordered_pair(I,J)
& in(ordered_pair(apply(C,I),apply(C,J)),B) ) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,cartesian_product2(A,A))
& F = E
& ? [K,L] :
( E = ordered_pair(K,L)
& in(ordered_pair(apply(C,K),apply(C,L)),B) ) ) ) ) ) ).
fof(s1_tarski__e6_22__wellord2__1,axiom,
! [A] :
( ! [B,C,D] :
( ( B = C
& ordinal(C)
& B = D
& ordinal(D) )
=> C = D )
=> ? [B] :
! [C] :
( in(C,B)
<=> ? [D] :
( in(D,A)
& D = C
& ordinal(C) ) ) ) ).
fof(s1_tarski__e6_27__finset_1__1,axiom,
! [A,B,C] :
( ( element(B,powerset(powerset(A)))
& relation(C)
& function(C) )
=> ( ! [D,E,F] :
( ( D = E
& in(relation_image(C,E),B)
& D = F
& in(relation_image(C,F),B) )
=> E = F )
=> ? [D] :
! [E] :
( in(E,D)
<=> ? [F] :
( in(F,powerset(relation_dom(C)))
& F = E
& in(relation_image(C,E),B) ) ) ) ) ).
fof(s1_tarski__e8_6__wellord2__1,axiom,
! [A,B] :
( ordinal(B)
=> ( ! [C,D,E] :
( ( C = D
& ? [F] :
( ordinal(F)
& D = F
& in(F,A) )
& C = E
& ? [G] :
( ordinal(G)
& E = G
& in(G,A) ) )
=> D = E )
=> ? [C] :
! [D] :
( in(D,C)
<=> ? [E] :
( in(E,succ(B))
& E = D
& ? [H] :
( ordinal(H)
& D = H
& in(H,A) ) ) ) ) ) ).
fof(s1_xboole_0__e10_24__wellord2__1,lemma,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ! [C] :
? [D] :
! [E] :
( in(E,D)
<=> ( in(E,cartesian_product2(A,C))
& ? [F,G] :
( ordered_pair(F,G) = E
& in(F,A)
& ? [H] :
( F = H
& in(G,H)
& ! [I] :
( in(I,H)
=> in(ordered_pair(G,I),B) ) ) ) ) ) ) ).
fof(s1_xboole_0__e11_2_1__waybel_0__1,lemma,
! [A,B,C] :
( ( ~ empty_carrier(A)
& transitive_relstr(A)
& rel_str(A)
& element(B,powerset(the_carrier(A)))
& finite(C)
& element(C,powerset(B)) )
=> ? [D] :
! [E] :
( in(E,D)
<=> ( in(E,powerset(C))
& ? [F] :
( F = E
& ? [G] :
( element(G,the_carrier(A))
& in(G,B)
& relstr_set_smaller(A,F,G) ) ) ) ) ) ).
fof(s1_xboole_0__e16_22__wellord2__1,lemma,
! [A,B] :
? [C] :
! [D] :
( in(D,C)
<=> ( in(D,cartesian_product2(A,B))
& ? [E,F] :
( ordered_pair(E,F) = D
& in(E,A)
& F = singleton(E) ) ) ) ).
fof(s1_xboole_0__e18_27__finset_1__1,lemma,
! [A] :
( ordinal(A)
=> ? [B] :
! [C] :
( in(C,B)
<=> ( in(C,succ(A))
& ? [D] :
( ordinal(D)
& C = 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 ) ) ) ) ) ) ) ) ) ).
fof(s1_xboole_0__e1_40__pre_topc__1,lemma,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> ? [C] :
! [D] :
( in(D,C)
<=> ( in(D,powerset(the_carrier(A)))
& ? [E] :
( element(E,powerset(the_carrier(A)))
& E = D
& closed_subset(E,A)
& subset(B,D) ) ) ) ) ).
fof(s1_xboole_0__e2_37_1_1__pre_topc__1,lemma,
! [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(s1_xboole_0__e4_27_3_1__finset_1,lemma,
! [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__e4_7_1__tops_2__1,lemma,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ! [C] :
? [D] :
! [E] :
( in(E,D)
<=> ( in(E,cartesian_product2(complements_of_subsets(the_carrier(A),B),C))
& ? [F,G] :
( ordered_pair(F,G) = E
& in(F,complements_of_subsets(the_carrier(A),B))
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = F
=> G = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ).
fof(s1_xboole_0__e4_7_2__tops_2__1,lemma,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ! [C] :
? [D] :
! [E] :
( in(E,D)
<=> ( in(E,cartesian_product2(B,C))
& ? [F,G] :
( ordered_pair(F,G) = E
& in(F,B)
& ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = F
=> G = subset_complement(the_carrier(A),H) ) ) ) ) ) ) ).
fof(s1_xboole_0__e6_21__wellord2__1,lemma,
! [A,B,C] :
( ( relation(B)
& relation(C)
& function(C) )
=> ? [D] :
! [E] :
( in(E,D)
<=> ( in(E,cartesian_product2(A,A))
& ? [F,G] :
( E = ordered_pair(F,G)
& in(ordered_pair(apply(C,F),apply(C,G)),B) ) ) ) ) ).
fof(s1_xboole_0__e6_22__wellord2,lemma,
! [A] :
? [B] :
! [C] :
( in(C,B)
<=> ( in(C,A)
& ordinal(C) ) ) ).
fof(s1_xboole_0__e6_27__finset_1,lemma,
! [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(s1_xboole_0__e8_6__wellord2__1,lemma,
! [A,B] :
( ordinal(B)
=> ? [C] :
! [D] :
( in(D,C)
<=> ( in(D,succ(B))
& ? [E] :
( ordinal(E)
& D = E
& in(E,A) ) ) ) ) ).
fof(s2_finset_1__e11_2_1__waybel_0,lemma,
! [A,B,C] :
( ( ~ empty_carrier(A)
& transitive_relstr(A)
& rel_str(A)
& element(B,powerset(the_carrier(A)))
& finite(C)
& element(C,powerset(B)) )
=> ( ( finite(C)
& ? [D] :
( element(D,the_carrier(A))
& in(D,B)
& relstr_set_smaller(A,empty_set,D) )
& ! [E,F] :
( ( in(E,C)
& subset(F,C)
& ? [G] :
( element(G,the_carrier(A))
& in(G,B)
& relstr_set_smaller(A,F,G) ) )
=> ? [H] :
( element(H,the_carrier(A))
& in(H,B)
& relstr_set_smaller(A,set_union2(F,singleton(E)),H) ) ) )
=> ? [I] :
( element(I,the_carrier(A))
& in(I,B)
& relstr_set_smaller(A,C,I) ) ) ) ).
fof(s2_funct_1__e10_24__wellord2,lemma,
! [A,B] :
( ( ~ empty(A)
& relation(B) )
=> ( ( ! [C,D,E] :
( ( in(C,A)
& ? [F] :
( C = F
& in(D,F)
& ! [G] :
( in(G,F)
=> in(ordered_pair(D,G),B) ) )
& ? [H] :
( C = H
& in(E,H)
& ! [I] :
( in(I,H)
=> in(ordered_pair(E,I),B) ) ) )
=> D = E )
& ! [C] :
~ ( in(C,A)
& ! [D] :
~ ? [J] :
( C = J
& in(D,J)
& ! [K] :
( in(K,J)
=> in(ordered_pair(D,K),B) ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = A
& ! [D] :
( in(D,A)
=> ? [L] :
( D = L
& in(apply(C,D),L)
& ! [M] :
( in(M,L)
=> in(ordered_pair(apply(C,D),M),B) ) ) ) ) ) ) ).
fof(s2_funct_1__e16_22__wellord2__1,lemma,
! [A] :
( ( ! [B,C,D] :
( ( in(B,A)
& C = singleton(B)
& D = singleton(B) )
=> C = D )
& ! [B] :
~ ( in(B,A)
& ! [C] : C != singleton(B) ) )
=> ? [B] :
( relation(B)
& function(B)
& relation_dom(B) = A
& ! [C] :
( in(C,A)
=> apply(B,C) = singleton(C) ) ) ) ).
fof(s2_funct_1__e4_7_1__tops_2,lemma,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ( ! [C,D,E] :
( ( in(C,complements_of_subsets(the_carrier(A),B))
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
& ! [C] :
~ ( in(C,complements_of_subsets(the_carrier(A),B))
& ! [D] :
~ ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = C
=> D = subset_complement(the_carrier(A),H) ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = complements_of_subsets(the_carrier(A),B)
& ! [D] :
( in(D,complements_of_subsets(the_carrier(A),B))
=> ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = D
=> apply(C,D) = subset_complement(the_carrier(A),I) ) ) ) ) ) ) ).
fof(s2_funct_1__e4_7_2__tops_2,lemma,
! [A,B] :
( ( one_sorted_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ( ( ! [C,D,E] :
( ( in(C,B)
& ! [F] :
( element(F,powerset(the_carrier(A)))
=> ( F = C
=> D = subset_complement(the_carrier(A),F) ) )
& ! [G] :
( element(G,powerset(the_carrier(A)))
=> ( G = C
=> E = subset_complement(the_carrier(A),G) ) ) )
=> D = E )
& ! [C] :
~ ( in(C,B)
& ! [D] :
~ ! [H] :
( element(H,powerset(the_carrier(A)))
=> ( H = C
=> D = subset_complement(the_carrier(A),H) ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = B
& ! [D] :
( in(D,B)
=> ! [I] :
( element(I,powerset(the_carrier(A)))
=> ( I = D
=> apply(C,D) = subset_complement(the_carrier(A),I) ) ) ) ) ) ) ).
fof(s2_ordinal1__e18_27__finset_1__1,lemma,
( ! [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(s3_funct_1__e16_22__wellord2,lemma,
! [A] :
? [B] :
( relation(B)
& function(B)
& relation_dom(B) = A
& ! [C] :
( in(C,A)
=> apply(B,C) = singleton(C) ) ) ).
fof(s3_subset_1__e1_40__pre_topc,lemma,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(the_carrier(A))) )
=> ? [C] :
( element(C,powerset(powerset(the_carrier(A))))
& ! [D] :
( element(D,powerset(the_carrier(A)))
=> ( in(D,C)
<=> ? [E] :
( element(E,powerset(the_carrier(A)))
& E = D
& closed_subset(E,A)
& subset(B,D) ) ) ) ) ) ).
fof(s3_subset_1__e2_37_1_1__pre_topc,lemma,
! [A,B] :
( ( topological_space(A)
& top_str(A)
& element(B,powerset(powerset(the_carrier(A)))) )
=> ? [C] :
( element(C,powerset(powerset(the_carrier(A))))
& ! [D] :
( element(D,powerset(the_carrier(A)))
=> ( in(D,C)
<=> in(set_difference(cast_as_carrier_subset(A),D),B) ) ) ) ) ).
fof(symmetry_r1_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
=> disjoint(B,A) ) ).
fof(symmetry_r2_wellord2,axiom,
! [A,B] :
( equipotent(A,B)
=> equipotent(B,A) ) ).
fof(t106_zfmisc_1,lemma,
! [A,B,C,D] :
( in(ordered_pair(A,B),cartesian_product2(C,D))
<=> ( in(A,C)
& in(B,D) ) ) ).
fof(t10_ordinal1,lemma,
! [A] : in(A,succ(A)) ).
fof(t10_tops_2,lemma,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ( ~ ( B != empty_set
& complements_of_subsets(A,B) = empty_set )
& ~ ( complements_of_subsets(A,B) != empty_set
& B = empty_set ) ) ) ).
fof(t10_zfmisc_1,lemma,
! [A,B,C,D] :
~ ( unordered_pair(A,B) = unordered_pair(C,D)
& A != C
& A != D ) ).
fof(t115_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(A,relation_rng(relation_rng_restriction(B,C)))
<=> ( in(A,B)
& in(A,relation_rng(C)) ) ) ) ).
fof(t116_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_rng(relation_rng_restriction(A,B)),A) ) ).
fof(t117_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_rng_restriction(A,B),B) ) ).
fof(t118_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_rng(relation_rng_restriction(A,B)),relation_rng(B)) ) ).
fof(t118_zfmisc_1,lemma,
! [A,B,C] :
( subset(A,B)
=> ( subset(cartesian_product2(A,C),cartesian_product2(B,C))
& subset(cartesian_product2(C,A),cartesian_product2(C,B)) ) ) ).
fof(t119_relat_1,lemma,
! [A,B] :
( relation(B)
=> relation_rng(relation_rng_restriction(A,B)) = set_intersection2(relation_rng(B),A) ) ).
fof(t119_zfmisc_1,lemma,
! [A,B,C,D] :
( ( subset(A,B)
& subset(C,D) )
=> subset(cartesian_product2(A,C),cartesian_product2(B,D)) ) ).
fof(t11_tops_2,lemma,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ( B != empty_set
=> meet_of_subsets(A,complements_of_subsets(A,B)) = subset_complement(A,union_of_subsets(A,B)) ) ) ).
fof(t12_pre_topc,lemma,
! [A] :
( one_sorted_str(A)
=> cast_as_carrier_subset(A) = the_carrier(A) ) ).
fof(t12_relset_1,lemma,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> ( subset(relation_dom(C),A)
& subset(relation_rng(C),B) ) ) ).
fof(t12_tops_2,lemma,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ( B != empty_set
=> union_of_subsets(A,complements_of_subsets(A,B)) = subset_complement(A,meet_of_subsets(A,B)) ) ) ).
fof(t12_xboole_1,lemma,
! [A,B] :
( subset(A,B)
=> set_union2(A,B) = B ) ).
fof(t136_zfmisc_1,lemma,
! [A] :
? [B] :
( in(A,B)
& ! [C,D] :
( ( in(C,B)
& subset(D,C) )
=> in(D,B) )
& ! [C] :
( in(C,B)
=> in(powerset(C),B) )
& ! [C] :
~ ( subset(C,B)
& ~ are_equipotent(C,B)
& ~ in(C,B) ) ) ).
fof(t13_compts_1,lemma,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ( compact_top_space(A)
<=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ~ ( centered(B)
& closed_subsets(B,A)
& meet_of_subsets(the_carrier(A),B) = empty_set ) ) ) ) ).
fof(t13_finset_1,lemma,
! [A,B] :
( ( subset(A,B)
& finite(B) )
=> finite(A) ) ).
fof(t13_tops_2,lemma,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( finite(complements_of_subsets(the_carrier(A),B))
<=> finite(B) ) ) ) ).
fof(t140_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> relation_dom_restriction(relation_rng_restriction(A,C),B) = relation_rng_restriction(A,relation_dom_restriction(C,B)) ) ).
fof(t143_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(A,relation_image(C,B))
<=> ? [D] :
( in(D,relation_dom(C))
& in(ordered_pair(D,A),C)
& in(D,B) ) ) ) ).
fof(t144_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_image(B,A),relation_rng(B)) ) ).
fof(t145_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> subset(relation_image(B,relation_inverse_image(B,A)),A) ) ).
fof(t145_relat_1,lemma,
! [A,B] :
( relation(B)
=> relation_image(B,A) = relation_image(B,set_intersection2(relation_dom(B),A)) ) ).
fof(t146_funct_1,lemma,
! [A,B] :
( relation(B)
=> ( subset(A,relation_dom(B))
=> subset(A,relation_inverse_image(B,relation_image(B,A))) ) ) ).
fof(t146_relat_1,lemma,
! [A] :
( relation(A)
=> relation_image(A,relation_dom(A)) = relation_rng(A) ) ).
fof(t147_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( subset(A,relation_rng(B))
=> relation_image(B,relation_inverse_image(B,A)) = A ) ) ).
fof(t14_relset_1,lemma,
! [A,B,C,D] :
( relation_of2_as_subset(D,C,A)
=> ( subset(relation_rng(D),B)
=> relation_of2_as_subset(D,C,B) ) ) ).
fof(t15_finset_1,lemma,
! [A,B] :
( finite(A)
=> finite(set_intersection2(A,B)) ) ).
fof(t15_pre_topc,lemma,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> subset_intersection2(the_carrier(A),B,cast_as_carrier_subset(A)) = B ) ) ).
fof(t15_yellow_0,lemma,
! [A] :
( ( antisymmetric_relstr(A)
& rel_str(A) )
=> ! [B] :
( ex_sup_of_relstr_set(A,B)
<=> ? [C] :
( element(C,the_carrier(A))
& relstr_set_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_set_smaller(A,B,D)
=> related(A,C,D) ) ) ) ) ) ).
fof(t160_relat_1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> relation_rng(relation_composition(A,B)) = relation_image(B,relation_rng(A)) ) ) ).
fof(t166_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(A,relation_inverse_image(C,B))
<=> ? [D] :
( in(D,relation_rng(C))
& in(ordered_pair(A,D),C)
& in(D,B) ) ) ) ).
fof(t167_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_inverse_image(B,A),relation_dom(B)) ) ).
fof(t16_relset_1,lemma,
! [A,B,C,D] :
( relation_of2_as_subset(D,C,A)
=> ( subset(A,B)
=> relation_of2_as_subset(D,C,B) ) ) ).
fof(t16_tops_2,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( closed_subsets(B,A)
<=> open_subsets(complements_of_subsets(the_carrier(A),B),A) ) ) ) ).
fof(t16_wellord1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(A,relation_restriction(C,B))
<=> ( in(A,C)
& in(A,cartesian_product2(B,B)) ) ) ) ).
fof(t16_yellow_0,lemma,
! [A] :
( ( antisymmetric_relstr(A)
& rel_str(A) )
=> ! [B] :
( ex_inf_of_relstr_set(A,B)
<=> ? [C] :
( element(C,the_carrier(A))
& relstr_element_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_element_smaller(A,B,D)
=> related(A,D,C) ) ) ) ) ) ).
fof(t174_relat_1,lemma,
! [A,B] :
( relation(B)
=> ~ ( A != empty_set
& subset(A,relation_rng(B))
& relation_inverse_image(B,A) = empty_set ) ) ).
fof(t178_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( subset(A,B)
=> subset(relation_inverse_image(C,A),relation_inverse_image(C,B)) ) ) ).
fof(t17_finset_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( finite(A)
=> finite(relation_image(B,A)) ) ) ).
fof(t17_pre_topc,lemma,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> subset_complement(the_carrier(A),B) = subset_difference(the_carrier(A),cast_as_carrier_subset(A),B) ) ) ).
fof(t17_tops_2,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( open_subsets(B,A)
<=> closed_subsets(complements_of_subsets(the_carrier(A),B),A) ) ) ) ).
fof(t17_wellord1,lemma,
! [A,B] :
( relation(B)
=> relation_restriction(B,A) = relation_dom_restriction(relation_rng_restriction(A,B),A) ) ).
fof(t17_xboole_1,lemma,
! [A,B] : subset(set_intersection2(A,B),A) ).
fof(t18_finset_1,lemma,
! [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(t18_wellord1,lemma,
! [A,B] :
( relation(B)
=> relation_restriction(B,A) = relation_rng_restriction(A,relation_dom_restriction(B,A)) ) ).
fof(t19_wellord1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(A,relation_field(relation_restriction(C,B)))
=> ( in(A,relation_field(C))
& in(A,B) ) ) ) ).
fof(t19_xboole_1,lemma,
! [A,B,C] :
( ( subset(A,B)
& subset(A,C) )
=> subset(A,set_intersection2(B,C)) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = A ).
fof(t1_lattice3,lemma,
! [A,B] :
( element(B,the_carrier(boole_lattice(A)))
=> ! [C] :
( element(C,the_carrier(boole_lattice(A)))
=> ( join(boole_lattice(A),B,C) = set_union2(B,C)
& meet(boole_lattice(A),B,C) = set_intersection2(B,C) ) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t1_waybel_0,conjecture,
! [A] :
( ( ~ empty_carrier(A)
& transitive_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( ~ empty(B)
& directed_subset(B,A) )
<=> ! [C] :
( ( finite(C)
& element(C,powerset(B)) )
=> ? [D] :
( element(D,the_carrier(A))
& in(D,B)
& relstr_set_smaller(A,C,D) ) ) ) ) ) ).
fof(t1_xboole_1,lemma,
! [A,B,C] :
( ( subset(A,B)
& subset(B,C) )
=> subset(A,C) ) ).
fof(t1_zfmisc_1,lemma,
powerset(empty_set) = singleton(empty_set) ).
fof(t20_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(ordered_pair(A,B),C)
=> ( in(A,relation_dom(C))
& in(B,relation_rng(C)) ) ) ) ).
fof(t20_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( subset(relation_field(relation_restriction(B,A)),relation_field(B))
& subset(relation_field(relation_restriction(B,A)),A) ) ) ).
fof(t21_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( in(A,relation_dom(relation_composition(C,B)))
<=> ( in(A,relation_dom(C))
& in(apply(C,A),relation_dom(B)) ) ) ) ) ).
fof(t21_funct_2,lemma,
! [A,B,C,D] :
( ( function(D)
& quasi_total(D,A,B)
& relation_of2_as_subset(D,A,B) )
=> ! [E] :
( ( relation(E)
& function(E) )
=> ( in(C,A)
=> ( B = empty_set
| apply(relation_composition(D,E),C) = apply(E,apply(D,C)) ) ) ) ) ).
fof(t21_ordinal1,lemma,
! [A] :
( epsilon_transitive(A)
=> ! [B] :
( ordinal(B)
=> ( proper_subset(A,B)
=> in(A,B) ) ) ) ).
fof(t21_relat_1,lemma,
! [A] :
( relation(A)
=> subset(A,cartesian_product2(relation_dom(A),relation_rng(A))) ) ).
fof(t21_wellord1,lemma,
! [A,B,C] :
( relation(C)
=> subset(fiber(relation_restriction(C,A),B),fiber(C,B)) ) ).
fof(t22_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( in(A,relation_dom(relation_composition(C,B)))
=> apply(relation_composition(C,B),A) = apply(B,apply(C,A)) ) ) ) ).
fof(t22_pre_topc,lemma,
! [A] :
( one_sorted_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> subset_difference(the_carrier(A),cast_as_carrier_subset(A),subset_difference(the_carrier(A),cast_as_carrier_subset(A),B)) = B ) ) ).
fof(t22_relset_1,lemma,
! [A,B,C] :
( relation_of2_as_subset(C,B,A)
=> ( ! [D] :
~ ( in(D,B)
& ! [E] : ~ in(ordered_pair(D,E),C) )
<=> relation_dom_as_subset(B,A,C) = B ) ) ).
fof(t22_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( reflexive(B)
=> reflexive(relation_restriction(B,A)) ) ) ).
fof(t23_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( in(A,relation_dom(B))
=> apply(relation_composition(B,C),A) = apply(C,apply(B,A)) ) ) ) ).
fof(t23_lattices,lemma,
! [A] :
( ( ~ empty_carrier(A)
& meet_commutative(A)
& meet_absorbing(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> below(A,meet_commut(A,B,C),B) ) ) ) ).
fof(t23_ordinal1,lemma,
! [A,B] :
( ordinal(B)
=> ( in(A,B)
=> ordinal(A) ) ) ).
fof(t23_relset_1,lemma,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> ( ! [D] :
~ ( in(D,B)
& ! [E] : ~ in(ordered_pair(E,D),C) )
<=> relation_rng_as_subset(A,B,C) = B ) ) ).
fof(t23_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( connected(B)
=> connected(relation_restriction(B,A)) ) ) ).
fof(t24_ordinal1,lemma,
! [A] :
( ordinal(A)
=> ! [B] :
( ordinal(B)
=> ~ ( ~ in(A,B)
& A != B
& ~ in(B,A) ) ) ) ).
fof(t24_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( transitive(B)
=> transitive(relation_restriction(B,A)) ) ) ).
fof(t25_orders_2,lemma,
! [A] :
( ( antisymmetric_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( ( related(A,B,C)
& related(A,C,B) )
=> B = C ) ) ) ) ).
fof(t25_relat_1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( subset(A,B)
=> ( subset(relation_dom(A),relation_dom(B))
& subset(relation_rng(A),relation_rng(B)) ) ) ) ) ).
fof(t25_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( antisymmetric(B)
=> antisymmetric(relation_restriction(B,A)) ) ) ).
fof(t25_wellord2,lemma,
! [A,B] :
( relation(B)
=> ( well_orders(B,A)
=> ( relation_field(relation_restriction(B,A)) = A
& well_ordering(relation_restriction(B,A)) ) ) ) ).
fof(t26_finset_1,lemma,
! [A] :
( ( relation(A)
& function(A) )
=> ( finite(relation_dom(A))
=> finite(relation_rng(A)) ) ) ).
fof(t26_lattices,lemma,
! [A] :
( ( ~ empty_carrier(A)
& join_commutative(A)
& join_semilatt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( ( below(A,B,C)
& below(A,C,B) )
=> B = C ) ) ) ) ).
fof(t26_orders_2,lemma,
! [A] :
( ( transitive_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ! [D] :
( element(D,the_carrier(A))
=> ( ( related(A,B,C)
& related(A,C,D) )
=> related(A,B,D) ) ) ) ) ) ).
fof(t26_wellord2,lemma,
! [A] :
? [B] :
( relation(B)
& well_orders(B,A) ) ).
fof(t26_xboole_1,lemma,
! [A,B,C] :
( subset(A,B)
=> subset(set_intersection2(A,C),set_intersection2(B,C)) ) ).
fof(t28_lattice3,lemma,
! [A,B] :
( ( ~ empty_carrier(B)
& lattice(B)
& latt_str(B) )
=> ! [C] :
( element(C,the_carrier(B))
=> ( latt_set_smaller(B,C,A)
<=> relstr_element_smaller(poset_of_lattice(B),A,cast_to_el_of_LattPOSet(B,C)) ) ) ) ).
fof(t28_wellord2,lemma,
! [A] :
( ~ empty(A)
=> ~ ( ! [B] :
~ ( in(B,A)
& B = empty_set )
& ! [B] :
( ( relation(B)
& function(B) )
=> ~ ( relation_dom(B) = A
& ! [C] :
( in(C,A)
=> in(apply(B,C),C) ) ) ) ) ) ).
fof(t28_xboole_1,lemma,
! [A,B] :
( subset(A,B)
=> set_intersection2(A,B) = A ) ).
fof(t29_lattice3,lemma,
! [A,B] :
( ( ~ empty_carrier(B)
& lattice(B)
& latt_str(B) )
=> ! [C] :
( element(C,the_carrier(poset_of_lattice(B)))
=> ( relstr_element_smaller(poset_of_lattice(B),A,C)
<=> latt_set_smaller(B,cast_to_el_of_lattice(B,C),A) ) ) ) ).
fof(t29_tops_1,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( closed_subset(B,A)
<=> open_subset(subset_complement(the_carrier(A),B),A) ) ) ) ).
fof(t29_yellow_0,lemma,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A)
& latt_str(A) )
=> ! [B] :
( join_of_latt_set(A,B) = join_on_relstr(poset_of_lattice(A),B)
& meet_of_latt_set(A,B) = meet_on_relstr(poset_of_lattice(A),B) ) ) ).
fof(t2_boole,axiom,
! [A] : set_intersection2(A,empty_set) = empty_set ).
fof(t2_lattice3,lemma,
! [A,B] :
( element(B,the_carrier(boole_lattice(A)))
=> ! [C] :
( element(C,the_carrier(boole_lattice(A)))
=> ( below(boole_lattice(A),B,C)
<=> subset(B,C) ) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( in(C,A)
<=> in(C,B) )
=> A = B ) ).
fof(t2_wellord2,lemma,
! [A] : reflexive(inclusion_relation(A)) ).
fof(t2_xboole_1,lemma,
! [A] : subset(empty_set,A) ).
fof(t30_lattice3,lemma,
! [A,B] :
( ( ~ empty_carrier(B)
& lattice(B)
& latt_str(B) )
=> ! [C] :
( element(C,the_carrier(B))
=> ( latt_element_smaller(B,C,A)
<=> relstr_set_smaller(poset_of_lattice(B),A,cast_to_el_of_LattPOSet(B,C)) ) ) ) ).
fof(t30_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(ordered_pair(A,B),C)
=> ( in(A,relation_field(C))
& in(B,relation_field(C)) ) ) ) ).
fof(t30_tops_1,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( open_subset(B,A)
<=> closed_subset(subset_complement(the_carrier(A),B),A) ) ) ) ).
fof(t30_yellow_0,lemma,
! [A] :
( ( antisymmetric_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( ( ( B = join_on_relstr(A,C)
& ex_sup_of_relstr_set(A,C) )
=> ( relstr_set_smaller(A,C,B)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_set_smaller(A,C,D)
=> related(A,B,D) ) ) ) )
& ( ( relstr_set_smaller(A,C,B)
& ! [D] :
( element(D,the_carrier(A))
=> ( relstr_set_smaller(A,C,D)
=> related(A,B,D) ) ) )
=> ( B = join_on_relstr(A,C)
& ex_sup_of_relstr_set(A,C) ) ) ) ) ) ).
fof(t31_lattice3,lemma,
! [A,B] :
( ( ~ empty_carrier(B)
& lattice(B)
& latt_str(B) )
=> ! [C] :
( element(C,the_carrier(poset_of_lattice(B)))
=> ( relstr_set_smaller(poset_of_lattice(B),A,C)
<=> latt_element_smaller(B,cast_to_el_of_lattice(B,C),A) ) ) ) ).
fof(t31_ordinal1,lemma,
! [A] :
( ! [B] :
( in(B,A)
=> ( ordinal(B)
& subset(B,A) ) )
=> ordinal(A) ) ).
fof(t31_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( well_founded_relation(B)
=> well_founded_relation(relation_restriction(B,A)) ) ) ).
fof(t32_filter_1,lemma,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( in(ordered_pair_as_product_element(the_carrier(A),the_carrier(A),B,C),relation_of_lattice(A))
<=> below_refl(A,B,C) ) ) ) ) ).
fof(t32_ordinal1,lemma,
! [A,B] :
( ordinal(B)
=> ~ ( subset(A,B)
& A != empty_set
& ! [C] :
( ordinal(C)
=> ~ ( in(C,A)
& ! [D] :
( ordinal(D)
=> ( in(D,A)
=> ordinal_subset(C,D) ) ) ) ) ) ) ).
fof(t32_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( well_ordering(B)
=> well_ordering(relation_restriction(B,A)) ) ) ).
fof(t33_ordinal1,lemma,
! [A] :
( ordinal(A)
=> ! [B] :
( ordinal(B)
=> ( in(A,B)
<=> ordinal_subset(succ(A),B) ) ) ) ).
fof(t33_xboole_1,lemma,
! [A,B,C] :
( subset(A,B)
=> subset(set_difference(A,C),set_difference(B,C)) ) ).
fof(t33_zfmisc_1,lemma,
! [A,B,C,D] :
( ordered_pair(A,B) = ordered_pair(C,D)
=> ( A = C
& B = D ) ) ).
fof(t34_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( B = identity_relation(A)
<=> ( relation_dom(B) = A
& ! [C] :
( in(C,A)
=> apply(B,C) = C ) ) ) ) ).
fof(t34_lattice3,lemma,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( B = meet_of_latt_set(A,C)
<=> ( latt_set_smaller(A,B,C)
& ! [D] :
( element(D,the_carrier(A))
=> ( latt_set_smaller(A,D,C)
=> below_refl(A,D,B) ) ) ) ) ) ) ).
fof(t35_funct_1,lemma,
! [A,B] :
( in(B,A)
=> apply(identity_relation(A),B) = B ) ).
fof(t36_xboole_1,lemma,
! [A,B] : subset(set_difference(A,B),A) ).
fof(t37_relat_1,lemma,
! [A] :
( relation(A)
=> ( relation_rng(A) = relation_dom(relation_inverse(A))
& relation_dom(A) = relation_rng(relation_inverse(A)) ) ) ).
fof(t37_xboole_1,lemma,
! [A,B] :
( set_difference(A,B) = empty_set
<=> subset(A,B) ) ).
fof(t37_zfmisc_1,lemma,
! [A,B] :
( subset(singleton(A),B)
<=> in(A,B) ) ).
fof(t38_zfmisc_1,lemma,
! [A,B,C] :
( subset(unordered_pair(A,B),C)
<=> ( in(A,C)
& in(B,C) ) ) ).
fof(t39_wellord1,lemma,
! [A,B] :
( relation(B)
=> ( ( well_ordering(B)
& subset(A,relation_field(B)) )
=> relation_field(relation_restriction(B,A)) = A ) ) ).
fof(t39_xboole_1,lemma,
! [A,B] : set_union2(A,set_difference(B,A)) = set_union2(A,B) ).
fof(t39_zfmisc_1,lemma,
! [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_lattice3,lemma,
! [A] :
( lower_bounded_semilattstr(boole_lattice(A))
& bottom_of_semilattstr(boole_lattice(A)) = empty_set ) ).
fof(t3_ordinal1,lemma,
! [A,B,C] :
~ ( in(A,B)
& in(B,C)
& in(C,A) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t3_wellord2,lemma,
! [A] : transitive(inclusion_relation(A)) ).
fof(t3_xboole_0,lemma,
! [A,B] :
( ~ ( ~ disjoint(A,B)
& ! [C] :
~ ( in(C,A)
& in(C,B) ) )
& ~ ( ? [C] :
( in(C,A)
& in(C,B) )
& disjoint(A,B) ) ) ).
fof(t3_xboole_1,lemma,
! [A] :
( subset(A,empty_set)
=> A = empty_set ) ).
fof(t40_xboole_1,lemma,
! [A,B] : set_difference(set_union2(A,B),B) = set_difference(A,B) ).
fof(t41_ordinal1,lemma,
! [A] :
( ordinal(A)
=> ( being_limit_ordinal(A)
<=> ! [B] :
( ordinal(B)
=> ( in(B,A)
=> in(succ(B),A) ) ) ) ) ).
fof(t42_ordinal1,lemma,
! [A] :
( ordinal(A)
=> ( ~ ( ~ being_limit_ordinal(A)
& ! [B] :
( ordinal(B)
=> A != succ(B) ) )
& ~ ( ? [B] :
( ordinal(B)
& A = succ(B) )
& being_limit_ordinal(A) ) ) ) ).
fof(t42_yellow_0,lemma,
! [A] :
( ( ~ empty_carrier(A)
& antisymmetric_relstr(A)
& lower_bounded_relstr(A)
& rel_str(A) )
=> ( ex_sup_of_relstr_set(A,empty_set)
& ex_inf_of_relstr_set(A,the_carrier(A)) ) ) ).
fof(t43_subset_1,lemma,
! [A,B] :
( element(B,powerset(A))
=> ! [C] :
( element(C,powerset(A))
=> ( disjoint(B,C)
<=> subset(B,subset_complement(A,C)) ) ) ) ).
fof(t44_pre_topc,lemma,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ( ! [C] :
( element(C,powerset(the_carrier(A)))
=> ( in(C,B)
=> closed_subset(C,A) ) )
=> closed_subset(meet_of_subsets(the_carrier(A),B),A) ) ) ) ).
fof(t44_relat_1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> subset(relation_dom(relation_composition(A,B)),relation_dom(A)) ) ) ).
fof(t44_tops_1,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> subset(interior(A,B),B) ) ) ).
fof(t44_yellow_0,lemma,
! [A] :
( ( ~ empty_carrier(A)
& antisymmetric_relstr(A)
& lower_bounded_relstr(A)
& rel_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> related(A,bottom_of_relstr(A),B) ) ) ).
fof(t45_pre_topc,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( in(C,the_carrier(A))
=> ( in(C,topstr_closure(A,B))
<=> ! [D] :
( element(D,powerset(the_carrier(A)))
=> ( ( closed_subset(D,A)
& subset(B,D) )
=> in(C,D) ) ) ) ) ) ) ).
fof(t45_relat_1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> subset(relation_rng(relation_composition(A,B)),relation_rng(B)) ) ) ).
fof(t45_xboole_1,lemma,
! [A,B] :
( subset(A,B)
=> B = set_union2(A,set_difference(B,A)) ) ).
fof(t46_funct_2,lemma,
! [A,B,C,D] :
( ( function(D)
& quasi_total(D,A,B)
& relation_of2_as_subset(D,A,B) )
=> ( B != empty_set
=> ! [E] :
( in(E,relation_inverse_image(D,C))
<=> ( in(E,A)
& in(apply(D,E),C) ) ) ) ) ).
fof(t46_pre_topc,lemma,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ? [C] :
( element(C,powerset(powerset(the_carrier(A))))
& ! [D] :
( element(D,powerset(the_carrier(A)))
=> ( in(D,C)
<=> ( closed_subset(D,A)
& subset(B,D) ) ) )
& topstr_closure(A,B) = meet_of_subsets(the_carrier(A),C) ) ) ) ).
fof(t46_relat_1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( subset(relation_rng(A),relation_dom(B))
=> relation_dom(relation_composition(A,B)) = relation_dom(A) ) ) ) ).
fof(t46_setfam_1,lemma,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ~ ( B != empty_set
& complements_of_subsets(A,B) = empty_set ) ) ).
fof(t46_zfmisc_1,lemma,
! [A,B] :
( in(A,B)
=> set_union2(singleton(A),B) = B ) ).
fof(t47_relat_1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( subset(relation_dom(A),relation_rng(B))
=> relation_rng(relation_composition(B,A)) = relation_rng(A) ) ) ) ).
fof(t47_setfam_1,lemma,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ( B != empty_set
=> subset_difference(A,cast_to_subset(A),union_of_subsets(A,B)) = meet_of_subsets(A,complements_of_subsets(A,B)) ) ) ).
fof(t48_pre_topc,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> subset(B,topstr_closure(A,B)) ) ) ).
fof(t48_setfam_1,lemma,
! [A,B] :
( element(B,powerset(powerset(A)))
=> ( B != empty_set
=> union_of_subsets(A,complements_of_subsets(A,B)) = subset_difference(A,cast_to_subset(A),meet_of_subsets(A,B)) ) ) ).
fof(t48_xboole_1,lemma,
! [A,B] : set_difference(A,set_difference(A,B)) = set_intersection2(A,B) ).
fof(t49_wellord1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( relation_isomorphism(A,B,C)
=> relation_isomorphism(B,A,function_inverse(C)) ) ) ) ) ).
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(t4_wellord2,lemma,
! [A] :
( ordinal(A)
=> connected(inclusion_relation(A)) ) ).
fof(t4_xboole_0,lemma,
! [A,B] :
( ~ ( ~ disjoint(A,B)
& ! [C] : ~ in(C,set_intersection2(A,B)) )
& ~ ( ? [C] : in(C,set_intersection2(A,B))
& disjoint(A,B) ) ) ).
fof(t50_lattice3,lemma,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& complete_latt_str(A)
& latt_str(A) )
=> ( ~ empty_carrier(A)
& lattice(A)
& lower_bounded_semilattstr(A)
& latt_str(A)
& bottom_of_semilattstr(A) = join_of_latt_set(A,empty_set) ) ) ).
fof(t50_subset_1,lemma,
! [A] :
( A != empty_set
=> ! [B] :
( element(B,powerset(A))
=> ! [C] :
( element(C,A)
=> ( ~ in(C,B)
=> in(C,subset_complement(A,B)) ) ) ) ) ).
fof(t51_tops_1,lemma,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> open_subset(interior(A,B),A) ) ) ).
fof(t52_pre_topc,lemma,
! [A] :
( top_str(A)
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ( ( closed_subset(B,A)
=> topstr_closure(A,B) = B )
& ( ( topological_space(A)
& topstr_closure(A,B) = B )
=> closed_subset(B,A) ) ) ) ) ).
fof(t53_wellord1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( relation_isomorphism(A,B,C)
=> ( ( reflexive(A)
=> reflexive(B) )
& ( transitive(A)
=> transitive(B) )
& ( connected(A)
=> connected(B) )
& ( antisymmetric(A)
=> antisymmetric(B) )
& ( well_founded_relation(A)
=> well_founded_relation(B) ) ) ) ) ) ) ).
fof(t54_funct_1,lemma,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> ! [B] :
( ( relation(B)
& function(B) )
=> ( B = function_inverse(A)
<=> ( relation_dom(B) = relation_rng(A)
& ! [C,D] :
( ( ( in(C,relation_rng(A))
& D = apply(B,C) )
=> ( in(D,relation_dom(A))
& C = apply(A,D) ) )
& ( ( in(D,relation_dom(A))
& C = apply(A,D) )
=> ( in(C,relation_rng(A))
& D = apply(B,C) ) ) ) ) ) ) ) ) ).
fof(t54_subset_1,lemma,
! [A,B,C] :
( element(C,powerset(A))
=> ~ ( in(B,subset_complement(A,C))
& in(B,C) ) ) ).
fof(t54_wellord1,lemma,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( ( well_ordering(A)
& relation_isomorphism(A,B,C) )
=> well_ordering(B) ) ) ) ) ).
fof(t55_funct_1,lemma,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> ( relation_rng(A) = relation_dom(function_inverse(A))
& relation_dom(A) = relation_rng(function_inverse(A)) ) ) ) ).
fof(t55_tops_1,lemma,
! [A] :
( ( topological_space(A)
& top_str(A) )
=> ! [B] :
( top_str(B)
=> ! [C] :
( element(C,powerset(the_carrier(A)))
=> ! [D] :
( element(D,powerset(the_carrier(B)))
=> ( ( open_subset(D,B)
=> interior(B,D) = D )
& ( interior(A,C) = C
=> open_subset(C,A) ) ) ) ) ) ) ).
fof(t56_relat_1,lemma,
! [A] :
( relation(A)
=> ( ! [B,C] : ~ in(ordered_pair(B,C),A)
=> A = empty_set ) ) ).
fof(t57_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( ( one_to_one(B)
& in(A,relation_rng(B)) )
=> ( A = apply(B,apply(function_inverse(B),A))
& A = apply(relation_composition(function_inverse(B),B),A) ) ) ) ).
fof(t5_connsp_2,lemma,
! [A] :
( ( ~ empty_carrier(A)
& topological_space(A)
& top_str(A) )
=> ! [B] :
( element(B,powerset(the_carrier(A)))
=> ! [C] :
( element(C,the_carrier(A))
=> ( ( open_subset(B,A)
& in(C,B) )
=> point_neighbourhood(B,A,C) ) ) ) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t5_tex_2,lemma,
! [A,B] :
( element(B,powerset(A))
=> ( proper_element(B,powerset(A))
<=> B != A ) ) ).
fof(t5_tops_2,lemma,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( element(B,powerset(powerset(the_carrier(A))))
=> ~ ( is_a_cover_of_carrier(A,B)
& B = empty_set ) ) ) ).
fof(t5_wellord1,lemma,
! [A] :
( relation(A)
=> ( well_founded_relation(A)
<=> is_well_founded_in(A,relation_field(A)) ) ) ).
fof(t5_wellord2,lemma,
! [A] : antisymmetric(inclusion_relation(A)) ).
fof(t60_relat_1,lemma,
( relation_dom(empty_set) = empty_set
& relation_rng(empty_set) = empty_set ) ).
fof(t60_xboole_1,lemma,
! [A,B] :
~ ( subset(A,B)
& proper_subset(B,A) ) ).
fof(t60_yellow_0,lemma,
! [A] :
( rel_str(A)
=> ! [B] :
( subrelstr(B,A)
=> ! [C] :
( element(C,the_carrier(A))
=> ! [D] :
( element(D,the_carrier(A))
=> ! [E] :
( element(E,the_carrier(B))
=> ! [F] :
( element(F,the_carrier(B))
=> ( ( E = C
& F = D
& related(B,E,F) )
=> related(A,C,D) ) ) ) ) ) ) ) ).
fof(t61_yellow_0,lemma,
! [A] :
( rel_str(A)
=> ! [B] :
( ( full_subrelstr(B,A)
& subrelstr(B,A) )
=> ! [C] :
( element(C,the_carrier(A))
=> ! [D] :
( element(D,the_carrier(A))
=> ! [E] :
( element(E,the_carrier(B))
=> ! [F] :
( element(F,the_carrier(B))
=> ( ( E = C
& F = D
& related(A,C,D)
& in(E,the_carrier(B))
& in(F,the_carrier(B)) )
=> related(B,E,F) ) ) ) ) ) ) ) ).
fof(t62_funct_1,lemma,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
=> one_to_one(function_inverse(A)) ) ) ).
fof(t63_xboole_1,lemma,
! [A,B,C] :
( ( subset(A,B)
& disjoint(B,C) )
=> disjoint(A,C) ) ).
fof(t64_relat_1,lemma,
! [A] :
( relation(A)
=> ( ( relation_dom(A) = empty_set
| relation_rng(A) = empty_set )
=> A = empty_set ) ) ).
fof(t65_relat_1,lemma,
! [A] :
( relation(A)
=> ( relation_dom(A) = empty_set
<=> relation_rng(A) = empty_set ) ) ).
fof(t65_zfmisc_1,lemma,
! [A,B] :
( set_difference(A,singleton(B)) = A
<=> ~ in(B,A) ) ).
fof(t68_funct_1,lemma,
! [A,B] :
( ( relation(B)
& function(B) )
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( B = relation_dom_restriction(C,A)
<=> ( relation_dom(B) = set_intersection2(relation_dom(C),A)
& ! [D] :
( in(D,relation_dom(B))
=> apply(B,D) = apply(C,D) ) ) ) ) ) ).
fof(t69_enumset1,lemma,
! [A] : unordered_pair(A,A) = singleton(A) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t6_funct_2,lemma,
! [A,B,C,D] :
( ( function(D)
& quasi_total(D,A,B)
& relation_of2_as_subset(D,A,B) )
=> ( in(C,A)
=> ( B = empty_set
| in(apply(D,C),relation_rng(D)) ) ) ) ).
fof(t6_wellord2,lemma,
! [A] :
( ordinal(A)
=> well_founded_relation(inclusion_relation(A)) ) ).
fof(t6_yellow_0,lemma,
! [A] :
( rel_str(A)
=> ! [B] :
( element(B,the_carrier(A))
=> ( relstr_set_smaller(A,empty_set,B)
& relstr_element_smaller(A,empty_set,B) ) ) ) ).
fof(t6_zfmisc_1,lemma,
! [A,B] :
( subset(singleton(A),singleton(B))
=> A = B ) ).
fof(t70_funct_1,lemma,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(B,relation_dom(relation_dom_restriction(C,A)))
=> apply(relation_dom_restriction(C,A),B) = apply(C,B) ) ) ).
fof(t71_relat_1,lemma,
! [A] :
( relation_dom(identity_relation(A)) = A
& relation_rng(identity_relation(A)) = A ) ).
fof(t72_funct_1,lemma,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(B,A)
=> apply(relation_dom_restriction(C,A),B) = apply(C,B) ) ) ).
fof(t74_relat_1,lemma,
! [A,B,C,D] :
( relation(D)
=> ( in(ordered_pair(A,B),relation_composition(identity_relation(C),D))
<=> ( in(A,C)
& in(ordered_pair(A,B),D) ) ) ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t7_lattice3,lemma,
! [A] :
( ( ~ empty_carrier(A)
& lattice(A)
& latt_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> ! [C] :
( element(C,the_carrier(A))
=> ( below_refl(A,B,C)
<=> related_reflexive(poset_of_lattice(A),cast_to_el_of_LattPOSet(A,B),cast_to_el_of_LattPOSet(A,C)) ) ) ) ) ).
fof(t7_mcart_1,lemma,
! [A,B] :
( pair_first(ordered_pair(A,B)) = A
& pair_second(ordered_pair(A,B)) = B ) ).
fof(t7_tarski,axiom,
! [A,B] :
~ ( in(A,B)
& ! [C] :
~ ( in(C,B)
& ! [D] :
~ ( in(D,B)
& in(D,C) ) ) ) ).
fof(t7_wellord2,lemma,
! [A] :
( ordinal(A)
=> well_ordering(inclusion_relation(A)) ) ).
fof(t7_xboole_1,lemma,
! [A,B] : subset(A,set_union2(A,B)) ).
fof(t83_xboole_1,lemma,
! [A,B] :
( disjoint(A,B)
<=> set_difference(A,B) = A ) ).
fof(t86_relat_1,lemma,
! [A,B,C] :
( relation(C)
=> ( in(A,relation_dom(relation_dom_restriction(C,B)))
<=> ( in(A,B)
& in(A,relation_dom(C)) ) ) ) ).
fof(t88_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_dom_restriction(B,A),B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
fof(t8_funct_1,lemma,
! [A,B,C] :
( ( relation(C)
& function(C) )
=> ( in(ordered_pair(A,B),C)
<=> ( in(A,relation_dom(C))
& B = apply(C,A) ) ) ) ).
fof(t8_wellord1,lemma,
! [A] :
( relation(A)
=> ( well_orders(A,relation_field(A))
<=> well_ordering(A) ) ) ).
fof(t8_xboole_1,lemma,
! [A,B,C] :
( ( subset(A,B)
& subset(C,B) )
=> subset(set_union2(A,C),B) ) ).
fof(t8_zfmisc_1,lemma,
! [A,B,C] :
( singleton(A) = unordered_pair(B,C)
=> A = B ) ).
fof(t90_relat_1,lemma,
! [A,B] :
( relation(B)
=> relation_dom(relation_dom_restriction(B,A)) = set_intersection2(relation_dom(B),A) ) ).
fof(t91_tmap_1,lemma,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ! [B] :
( element(B,the_carrier(A))
=> apply_as_element(the_carrier(A),the_carrier(A),identity_on_carrier(A),B) = B ) ) ).
fof(t92_zfmisc_1,lemma,
! [A,B] :
( in(A,B)
=> subset(A,union(B)) ) ).
fof(t94_relat_1,lemma,
! [A,B] :
( relation(B)
=> relation_dom_restriction(B,A) = relation_composition(identity_relation(A),B) ) ).
fof(t99_relat_1,lemma,
! [A,B] :
( relation(B)
=> subset(relation_rng(relation_dom_restriction(B,A)),relation_rng(B)) ) ).
fof(t99_zfmisc_1,lemma,
! [A] : union(powerset(A)) = A ).
fof(t9_funct_2,lemma,
! [A,B,C,D] :
( ( function(D)
& quasi_total(D,A,B)
& relation_of2_as_subset(D,A,B) )
=> ( subset(B,C)
=> ( ( B = empty_set
& A != empty_set )
| ( function(D)
& quasi_total(D,A,C)
& relation_of2_as_subset(D,A,C) ) ) ) ) ).
fof(t9_tarski,axiom,
! [A] :
? [B] :
( in(A,B)
& ! [C,D] :
( ( in(C,B)
& subset(D,C) )
=> in(D,B) )
& ! [C] :
~ ( in(C,B)
& ! [D] :
~ ( in(D,B)
& ! [E] :
( subset(E,C)
=> in(E,D) ) ) )
& ! [C] :
~ ( subset(C,B)
& ~ are_equipotent(C,B)
& ~ in(C,B) ) ) ).
fof(t9_zfmisc_1,lemma,
! [A,B,C] :
( singleton(A) = unordered_pair(B,C)
=> B = C ) ).
%------------------------------------------------------------------------------