Article ORDERS_1, MML version 4.99.1005
:: ORDERS_1:modenot 1 => ORDERS_1:mode 1
definition
let a1 be non empty set;
assume not {} in a1;
mode Choice_Function of A1 -> Function-like quasi_total Relation of a1,union a1 means
for b1 being set
st b1 in a1
holds it . b1 in b1;
end;
:: ORDERS_1:dfs 1
definiens
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of a1,union a1;
To prove
a2 is Choice_Function of a1
it is sufficient to prove
thus not {} in a1;
thus for b1 being set
st b1 in a1
holds a2 . b1 in b1;
:: ORDERS_1:def 1
theorem
for b1 being non empty set
st not {} in b1
for b2 being Function-like quasi_total Relation of b1,union b1 holds
b2 is Choice_Function of b1
iff
for b3 being set
st b3 in b1
holds b2 . b3 in b3;
:: ORDERS_1:funcnot 1 => ORDERS_1:func 1
definition
let a1 be set;
func BOOL A1 -> set equals
(bool a1) \ {{}};
end;
:: ORDERS_1:def 2
theorem
for b1 being set holds
BOOL b1 = (bool b1) \ {{}};
:: ORDERS_1:funcreg 1
registration
let a1 be non empty set;
cluster BOOL a1 -> non empty;
end;
:: ORDERS_1:th 4
theorem
for b1 being non empty set holds
not {} in BOOL b1;
:: ORDERS_1:th 5
theorem
for b1, b2 being non empty set holds
b1 c= b2
iff
b1 in BOOL b2;
:: ORDERS_1:th 6
theorem
for b1, b2 being non empty set holds
b1 is Element of bool b2
iff
b1 in BOOL b2;
:: ORDERS_1:th 7
theorem
for b1 being non empty set holds
b1 in BOOL b1;
:: ORDERS_1:modenot 2
definition
let a1 be set;
mode Order of a1 is reflexive antisymmetric transitive total Relation of a1,a1;
end;
:: ORDERS_1:th 12
theorem
for b1, b2 being set
for b3 being reflexive antisymmetric transitive total Relation of b1,b1
st b2 in b1
holds [b2,b2] in b3;
:: ORDERS_1:th 13
theorem
for b1, b2, b3 being set
for b4 being reflexive antisymmetric transitive total Relation of b1,b1
st b2 in b1 & b3 in b1 & [b2,b3] in b4 & [b3,b2] in b4
holds b2 = b3;
:: ORDERS_1:th 14
theorem
for b1, b2, b3, b4 being set
for b5 being reflexive antisymmetric transitive total Relation of b1,b1
st b2 in b1 & b3 in b1 & b4 in b1 & [b2,b3] in b5 & [b3,b4] in b5
holds [b2,b4] in b5;
:: ORDERS_1:th 91
theorem
for b1 being set holds
ex b2 being set st
b2 <> {} & b2 in b1
iff
union b1 <> {};
:: ORDERS_1:th 92
theorem
for b1 being set
for b2 being Relation-like set holds
b2 is_strongly_connected_in b1
iff
b2 is_reflexive_in b1 & b2 is_connected_in b1;
:: ORDERS_1:th 93
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is_reflexive_in b1 & b2 c= b1
holds b3 is_reflexive_in b2;
:: ORDERS_1:th 94
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is_antisymmetric_in b1 & b2 c= b1
holds b3 is_antisymmetric_in b2;
:: ORDERS_1:th 95
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is_transitive_in b1 & b2 c= b1
holds b3 is_transitive_in b2;
:: ORDERS_1:th 96
theorem
for b1, b2 being set
for b3 being Relation-like set
st b3 is_strongly_connected_in b1 & b2 c= b1
holds b3 is_strongly_connected_in b2;
:: ORDERS_1:th 97
theorem
for b1 being set
for b2 being total Relation of b1,b1 holds
field b2 = b1;
:: ORDERS_1:th 98
theorem
for b1 being set
for b2 being Relation of b1,b1
st b2 is_reflexive_in b1
holds dom b2 = b1 & field b2 = b1;
:: ORDERS_1:th 99
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1 holds
dom b2 = b1 & rng b2 = b1;
:: ORDERS_1:th 100
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1 holds
field b2 = b1;
:: ORDERS_1:attrnot 1 => ORDERS_1:attr 1
definition
let a1 be Relation-like set;
attr a1 is being_quasi-order means
a1 is reflexive & a1 is transitive;
end;
:: ORDERS_1:dfs 3
definiens
let a1 be Relation-like set;
To prove
a1 is being_quasi-order
it is sufficient to prove
thus a1 is reflexive & a1 is transitive;
:: ORDERS_1:def 3
theorem
for b1 being Relation-like set holds
b1 is being_quasi-order
iff
b1 is reflexive & b1 is transitive;
:: ORDERS_1:attrnot 2 => ORDERS_1:attr 2
definition
let a1 be Relation-like set;
attr a1 is being_partial-order means
a1 is reflexive & a1 is transitive & a1 is antisymmetric;
end;
:: ORDERS_1:dfs 4
definiens
let a1 be Relation-like set;
To prove
a1 is being_partial-order
it is sufficient to prove
thus a1 is reflexive & a1 is transitive & a1 is antisymmetric;
:: ORDERS_1:def 4
theorem
for b1 being Relation-like set holds
b1 is being_partial-order
iff
b1 is reflexive & b1 is transitive & b1 is antisymmetric;
:: ORDERS_1:attrnot 3 => ORDERS_1:attr 3
definition
let a1 be Relation-like set;
attr a1 is being_linear-order means
a1 is reflexive & a1 is transitive & a1 is antisymmetric & a1 is connected;
end;
:: ORDERS_1:dfs 5
definiens
let a1 be Relation-like set;
To prove
a1 is being_linear-order
it is sufficient to prove
thus a1 is reflexive & a1 is transitive & a1 is antisymmetric & a1 is connected;
:: ORDERS_1:def 5
theorem
for b1 being Relation-like set holds
b1 is being_linear-order
iff
b1 is reflexive & b1 is transitive & b1 is antisymmetric & b1 is connected;
:: ORDERS_1:prednot 1 => ORDERS_1:attr 1
notation
let a1 be Relation-like set;
synonym a1 is_quasi-order for being_quasi-order;
end;
:: ORDERS_1:prednot 2 => ORDERS_1:attr 2
notation
let a1 be Relation-like set;
synonym a1 is_partial-order for being_partial-order;
end;
:: ORDERS_1:prednot 3 => ORDERS_1:attr 3
notation
let a1 be Relation-like set;
synonym a1 is_linear-order for being_linear-order;
end;
:: ORDERS_1:th 104
theorem
for b1 being Relation-like set
st b1 is being_quasi-order
holds b1 ~ is being_quasi-order;
:: ORDERS_1:th 105
theorem
for b1 being Relation-like set
st b1 is being_partial-order
holds b1 ~ is being_partial-order;
:: ORDERS_1:th 106
theorem
for b1 being Relation-like set
st b1 is being_linear-order
holds b1 ~ is being_linear-order;
:: ORDERS_1:th 107
theorem
for b1 being Relation-like set
st b1 is well-ordering
holds b1 is being_quasi-order & b1 is being_partial-order & b1 is being_linear-order;
:: ORDERS_1:th 108
theorem
for b1 being Relation-like set
st b1 is being_linear-order
holds b1 is being_quasi-order & b1 is being_partial-order;
:: ORDERS_1:th 109
theorem
for b1 being Relation-like set
st b1 is being_partial-order
holds b1 is being_quasi-order;
:: ORDERS_1:th 110
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1 holds
b2 is being_partial-order;
:: ORDERS_1:th 111
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1 holds
b2 is being_quasi-order;
:: ORDERS_1:th 112
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1
st b2 is connected
holds b2 is being_linear-order;
:: ORDERS_1:th 113
theorem
for b1 being Relation-like set
for b2 being set
st b1 is being_quasi-order
holds b1 |_2 b2 is being_quasi-order;
:: ORDERS_1:th 114
theorem
for b1 being Relation-like set
for b2 being set
st b1 is being_partial-order
holds b1 |_2 b2 is being_partial-order;
:: ORDERS_1:th 115
theorem
for b1 being Relation-like set
for b2 being set
st b1 is being_linear-order
holds b1 |_2 b2 is being_linear-order;
:: ORDERS_1:th 119
theorem
{} is being_quasi-order & {} is being_partial-order & {} is being_linear-order & {} is well-ordering;
:: ORDERS_1:th 120
theorem
for b1 being set holds
id b1 is being_quasi-order & id b1 is being_partial-order;
:: ORDERS_1:prednot 4 => ORDERS_1:pred 1
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 quasi_orders A2 means
a1 is_reflexive_in a2 & a1 is_transitive_in a2;
end;
:: ORDERS_1:dfs 6
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 quasi_orders a2
it is sufficient to prove
thus a1 is_reflexive_in a2 & a1 is_transitive_in a2;
:: ORDERS_1:def 6
theorem
for b1 being Relation-like set
for b2 being set holds
b1 quasi_orders b2
iff
b1 is_reflexive_in b2 & b1 is_transitive_in b2;
:: ORDERS_1:prednot 5 => ORDERS_1:pred 2
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 partially_orders A2 means
a1 is_reflexive_in a2 & a1 is_transitive_in a2 & a1 is_antisymmetric_in a2;
end;
:: ORDERS_1:dfs 7
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 partially_orders a2
it is sufficient to prove
thus a1 is_reflexive_in a2 & a1 is_transitive_in a2 & a1 is_antisymmetric_in a2;
:: ORDERS_1:def 7
theorem
for b1 being Relation-like set
for b2 being set holds
b1 partially_orders b2
iff
b1 is_reflexive_in b2 & b1 is_transitive_in b2 & b1 is_antisymmetric_in b2;
:: ORDERS_1:prednot 6 => ORDERS_1:pred 3
definition
let a1 be Relation-like set;
let a2 be set;
pred A1 linearly_orders A2 means
a1 is_reflexive_in a2 & a1 is_transitive_in a2 & a1 is_antisymmetric_in a2 & a1 is_connected_in a2;
end;
:: ORDERS_1:dfs 8
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a1 linearly_orders a2
it is sufficient to prove
thus a1 is_reflexive_in a2 & a1 is_transitive_in a2 & a1 is_antisymmetric_in a2 & a1 is_connected_in a2;
:: ORDERS_1:def 8
theorem
for b1 being Relation-like set
for b2 being set holds
b1 linearly_orders b2
iff
b1 is_reflexive_in b2 & b1 is_transitive_in b2 & b1 is_antisymmetric_in b2 & b1 is_connected_in b2;
:: ORDERS_1:th 124
theorem
for b1 being Relation-like set
for b2 being set
st b1 well_orders b2
holds b1 quasi_orders b2 & b1 partially_orders b2 & b1 linearly_orders b2;
:: ORDERS_1:th 125
theorem
for b1 being Relation-like set
for b2 being set
st b1 linearly_orders b2
holds b1 quasi_orders b2 & b1 partially_orders b2;
:: ORDERS_1:th 126
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2
holds b1 quasi_orders b2;
:: ORDERS_1:th 127
theorem
for b1 being Relation-like set
st b1 is being_quasi-order
holds b1 quasi_orders field b1;
:: ORDERS_1:th 128
theorem
for b1 being Relation-like set
for b2, b3 being set
st b1 quasi_orders b2 & b3 c= b2
holds b1 quasi_orders b3;
:: ORDERS_1:th 129
theorem
for b1 being Relation-like set
for b2 being set
st b1 quasi_orders b2
holds b1 |_2 b2 is being_quasi-order;
:: ORDERS_1:th 130
theorem
for b1 being Relation-like set
st b1 is being_partial-order
holds b1 partially_orders field b1;
:: ORDERS_1:th 131
theorem
for b1 being Relation-like set
for b2, b3 being set
st b1 partially_orders b2 & b3 c= b2
holds b1 partially_orders b3;
:: ORDERS_1:th 132
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2
holds b1 |_2 b2 is being_partial-order;
:: ORDERS_1:th 133
theorem
for b1 being Relation-like set
st b1 is being_linear-order
holds b1 linearly_orders field b1;
:: ORDERS_1:th 134
theorem
for b1 being Relation-like set
for b2, b3 being set
st b1 linearly_orders b2 & b3 c= b2
holds b1 linearly_orders b3;
:: ORDERS_1:th 135
theorem
for b1 being Relation-like set
for b2 being set
st b1 linearly_orders b2
holds b1 |_2 b2 is being_linear-order;
:: ORDERS_1:th 136
theorem
for b1 being Relation-like set
for b2 being set
st b1 quasi_orders b2
holds b1 ~ quasi_orders b2;
:: ORDERS_1:th 137
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2
holds b1 ~ partially_orders b2;
:: ORDERS_1:th 138
theorem
for b1 being Relation-like set
for b2 being set
st b1 linearly_orders b2
holds b1 ~ linearly_orders b2;
:: ORDERS_1:th 139
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1 holds
b2 quasi_orders b1;
:: ORDERS_1:th 140
theorem
for b1 being set
for b2 being reflexive antisymmetric transitive total Relation of b1,b1 holds
b2 partially_orders b1;
:: ORDERS_1:th 141
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2
holds b1 |_2 b2 is reflexive antisymmetric transitive total Relation of b2,b2;
:: ORDERS_1:th 142
theorem
for b1 being Relation-like set
for b2 being set
st b1 linearly_orders b2
holds b1 |_2 b2 is reflexive antisymmetric transitive total Relation of b2,b2;
:: ORDERS_1:th 143
theorem
for b1 being Relation-like set
for b2 being set
st b1 well_orders b2
holds b1 |_2 b2 is reflexive antisymmetric transitive total Relation of b2,b2;
:: ORDERS_1:th 146
theorem
for b1 being set holds
id b1 quasi_orders b1 & id b1 partially_orders b1;
:: ORDERS_1:prednot 7 => ORDERS_1:pred 4
definition
let a1 be Relation-like set;
let a2 be set;
pred A2 has_upper_Zorn_property_wrt A1 means
for b1 being set
st b1 c= a2 & a1 |_2 b1 is being_linear-order
holds ex b2 being set st
b2 in a2 &
(for b3 being set
st b3 in b1
holds [b3,b2] in a1);
end;
:: ORDERS_1:dfs 9
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a2 has_upper_Zorn_property_wrt a1
it is sufficient to prove
thus for b1 being set
st b1 c= a2 & a1 |_2 b1 is being_linear-order
holds ex b2 being set st
b2 in a2 &
(for b3 being set
st b3 in b1
holds [b3,b2] in a1);
:: ORDERS_1:def 9
theorem
for b1 being Relation-like set
for b2 being set holds
b2 has_upper_Zorn_property_wrt b1
iff
for b3 being set
st b3 c= b2 & b1 |_2 b3 is being_linear-order
holds ex b4 being set st
b4 in b2 &
(for b5 being set
st b5 in b3
holds [b5,b4] in b1);
:: ORDERS_1:prednot 8 => ORDERS_1:pred 5
definition
let a1 be Relation-like set;
let a2 be set;
pred A2 has_lower_Zorn_property_wrt A1 means
for b1 being set
st b1 c= a2 & a1 |_2 b1 is being_linear-order
holds ex b2 being set st
b2 in a2 &
(for b3 being set
st b3 in b1
holds [b2,b3] in a1);
end;
:: ORDERS_1:dfs 10
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a2 has_lower_Zorn_property_wrt a1
it is sufficient to prove
thus for b1 being set
st b1 c= a2 & a1 |_2 b1 is being_linear-order
holds ex b2 being set st
b2 in a2 &
(for b3 being set
st b3 in b1
holds [b2,b3] in a1);
:: ORDERS_1:def 10
theorem
for b1 being Relation-like set
for b2 being set holds
b2 has_lower_Zorn_property_wrt b1
iff
for b3 being set
st b3 c= b2 & b1 |_2 b3 is being_linear-order
holds ex b4 being set st
b4 in b2 &
(for b5 being set
st b5 in b3
holds [b4,b5] in b1);
:: ORDERS_1:th 149
theorem
for b1 being Relation-like set
for b2 being set
st b2 has_upper_Zorn_property_wrt b1
holds b2 <> {};
:: ORDERS_1:th 150
theorem
for b1 being Relation-like set
for b2 being set
st b2 has_lower_Zorn_property_wrt b1
holds b2 <> {};
:: ORDERS_1:th 151
theorem
for b1 being Relation-like set
for b2 being set holds
b2 has_upper_Zorn_property_wrt b1
iff
b2 has_lower_Zorn_property_wrt b1 ~;
:: ORDERS_1:th 152
theorem
for b1 being Relation-like set
for b2 being set holds
b2 has_upper_Zorn_property_wrt b1 ~
iff
b2 has_lower_Zorn_property_wrt b1;
:: ORDERS_1:prednot 9 => ORDERS_1:pred 6
definition
let a1 be Relation-like set;
let a2 be set;
pred A2 is_maximal_in A1 means
a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds not [a2,b1] in a1);
end;
:: ORDERS_1:dfs 11
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a2 is_maximal_in a1
it is sufficient to prove
thus a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds not [a2,b1] in a1);
:: ORDERS_1:def 11
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_maximal_in b1
iff
b2 in field b1 &
(for b3 being set
st b3 in field b1 & b3 <> b2
holds not [b2,b3] in b1);
:: ORDERS_1:prednot 10 => ORDERS_1:pred 7
definition
let a1 be Relation-like set;
let a2 be set;
pred A2 is_minimal_in A1 means
a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds not [b1,a2] in a1);
end;
:: ORDERS_1:dfs 12
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a2 is_minimal_in a1
it is sufficient to prove
thus a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds not [b1,a2] in a1);
:: ORDERS_1:def 12
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_minimal_in b1
iff
b2 in field b1 &
(for b3 being set
st b3 in field b1 & b3 <> b2
holds not [b3,b2] in b1);
:: ORDERS_1:prednot 11 => ORDERS_1:pred 8
definition
let a1 be Relation-like set;
let a2 be set;
pred A2 is_superior_of A1 means
a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds [b1,a2] in a1);
end;
:: ORDERS_1:dfs 13
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a2 is_superior_of a1
it is sufficient to prove
thus a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds [b1,a2] in a1);
:: ORDERS_1:def 13
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_superior_of b1
iff
b2 in field b1 &
(for b3 being set
st b3 in field b1 & b3 <> b2
holds [b3,b2] in b1);
:: ORDERS_1:prednot 12 => ORDERS_1:pred 9
definition
let a1 be Relation-like set;
let a2 be set;
pred A2 is_inferior_of A1 means
a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds [a2,b1] in a1);
end;
:: ORDERS_1:dfs 14
definiens
let a1 be Relation-like set;
let a2 be set;
To prove
a2 is_inferior_of a1
it is sufficient to prove
thus a2 in field a1 &
(for b1 being set
st b1 in field a1 & b1 <> a2
holds [a2,b1] in a1);
:: ORDERS_1:def 14
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_inferior_of b1
iff
b2 in field b1 &
(for b3 being set
st b3 in field b1 & b3 <> b2
holds [b2,b3] in b1);
:: ORDERS_1:th 157
theorem
for b1 being Relation-like set
for b2 being set
st b2 is_inferior_of b1 & b1 is antisymmetric
holds b2 is_minimal_in b1;
:: ORDERS_1:th 158
theorem
for b1 being Relation-like set
for b2 being set
st b2 is_superior_of b1 & b1 is antisymmetric
holds b2 is_maximal_in b1;
:: ORDERS_1:th 159
theorem
for b1 being Relation-like set
for b2 being set
st b2 is_minimal_in b1 & b1 is connected
holds b2 is_inferior_of b1;
:: ORDERS_1:th 160
theorem
for b1 being Relation-like set
for b2 being set
st b2 is_maximal_in b1 & b1 is connected
holds b2 is_superior_of b1;
:: ORDERS_1:th 161
theorem
for b1 being Relation-like set
for b2, b3 being set
st b2 in b3 & b2 is_superior_of b1 & b3 c= field b1 & b1 is reflexive
holds b3 has_upper_Zorn_property_wrt b1;
:: ORDERS_1:th 162
theorem
for b1 being Relation-like set
for b2, b3 being set
st b2 in b3 & b2 is_inferior_of b1 & b3 c= field b1 & b1 is reflexive
holds b3 has_lower_Zorn_property_wrt b1;
:: ORDERS_1:th 163
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_minimal_in b1
iff
b2 is_maximal_in b1 ~;
:: ORDERS_1:th 164
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_minimal_in b1 ~
iff
b2 is_maximal_in b1;
:: ORDERS_1:th 165
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_inferior_of b1
iff
b2 is_superior_of b1 ~;
:: ORDERS_1:th 166
theorem
for b1 being Relation-like set
for b2 being set holds
b2 is_inferior_of b1 ~
iff
b2 is_superior_of b1;
:: ORDERS_1:th 173
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2 & field b1 = b2 & b2 has_upper_Zorn_property_wrt b1
holds ex b3 being set st
b3 is_maximal_in b1;
:: ORDERS_1:th 174
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2 & field b1 = b2 & b2 has_lower_Zorn_property_wrt b1
holds ex b3 being set st
b3 is_minimal_in b1;
:: ORDERS_1:th 175
theorem
for b1 being set
st b1 <> {} &
(for b2 being set
st b2 c= b1 & b2 is c=-linear
holds ex b3 being set st
b3 in b1 &
(for b4 being set
st b4 in b2
holds b4 c= b3))
holds ex b2 being set st
b2 in b1 &
(for b3 being set
st b3 in b1 & b3 <> b2
holds not b2 c= b3);
:: ORDERS_1:th 176
theorem
for b1 being set
st b1 <> {} &
(for b2 being set
st b2 c= b1 & b2 is c=-linear
holds ex b3 being set st
b3 in b1 &
(for b4 being set
st b4 in b2
holds b3 c= b4))
holds ex b2 being set st
b2 in b1 &
(for b3 being set
st b3 in b1 & b3 <> b2
holds not b3 c= b2);
:: ORDERS_1:th 177
theorem
for b1 being set
st b1 <> {} &
(for b2 being set
st b2 <> {} & b2 c= b1 & b2 is c=-linear
holds union b2 in b1)
holds ex b2 being set st
b2 in b1 &
(for b3 being set
st b3 in b1 & b3 <> b2
holds not b2 c= b3);
:: ORDERS_1:th 178
theorem
for b1 being set
st b1 <> {} &
(for b2 being set
st b2 <> {} & b2 c= b1 & b2 is c=-linear
holds meet b2 in b1)
holds ex b2 being set st
b2 in b1 &
(for b3 being set
st b3 in b1 & b3 <> b2
holds not b3 c= b2);
:: ORDERS_1:sch 1
scheme ORDERS_1:sch 1
{F1 -> non empty set}:
ex b1 being Element of F1() st
for b2 being Element of F1()
st b1 <> b2
holds not (P1[b1, b2])
provided
for b1 being Element of F1() holds
P1[b1, b1]
and
for b1, b2 being Element of F1()
st P1[b1, b2] & P1[b2, b1]
holds b1 = b2
and
for b1, b2, b3 being Element of F1()
st P1[b1, b2] & P1[b2, b3]
holds P1[b1, b3]
and
for b1 being set
st b1 c= F1() &
(for b2, b3 being Element of F1()
st b2 in b1 & b3 in b1 & not (P1[b2, b3])
holds P1[b3, b2])
holds ex b2 being Element of F1() st
for b3 being Element of F1()
st b3 in b1
holds P1[b3, b2];
:: ORDERS_1:sch 2
scheme ORDERS_1:sch 2
{F1 -> non empty set}:
ex b1 being Element of F1() st
for b2 being Element of F1()
st b1 <> b2
holds not (P1[b2, b1])
provided
for b1 being Element of F1() holds
P1[b1, b1]
and
for b1, b2 being Element of F1()
st P1[b1, b2] & P1[b2, b1]
holds b1 = b2
and
for b1, b2, b3 being Element of F1()
st P1[b1, b2] & P1[b2, b3]
holds P1[b1, b3]
and
for b1 being set
st b1 c= F1() &
(for b2, b3 being Element of F1()
st b2 in b1 & b3 in b1 & not (P1[b2, b3])
holds P1[b3, b2])
holds ex b2 being Element of F1() st
for b3 being Element of F1()
st b3 in b1
holds P1[b2, b3];
:: ORDERS_1:th 179
theorem
for b1 being Relation-like set
for b2 being set
st b1 partially_orders b2 & field b1 = b2
holds ex b3 being Relation-like set st
b1 c= b3 & b3 linearly_orders b2 & field b3 = b2;
:: ORDERS_1:th 180
theorem
for b1 being Relation-like set holds
b1 c= [:field b1,field b1:];
:: ORDERS_1:th 181
theorem
for b1 being Relation-like set
for b2 being set
st b1 is reflexive & b2 c= field b1
holds field (b1 |_2 b2) = b2;
:: ORDERS_1:th 182
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_reflexive_in b2
holds b1 |_2 b2 is reflexive;
:: ORDERS_1:th 183
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_transitive_in b2
holds b1 |_2 b2 is transitive;
:: ORDERS_1:th 184
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_antisymmetric_in b2
holds b1 |_2 b2 is antisymmetric;
:: ORDERS_1:th 185
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_connected_in b2
holds b1 |_2 b2 is connected;
:: ORDERS_1:th 186
theorem
for b1 being Relation-like set
for b2, b3 being set
st b1 is_connected_in b2 & b3 c= b2
holds b1 is_connected_in b3;
:: ORDERS_1:th 187
theorem
for b1 being Relation-like set
for b2, b3 being set
st b1 well_orders b2 & b3 c= b2
holds b1 well_orders b3;
:: ORDERS_1:th 188
theorem
for b1 being Relation-like set
st b1 is connected
holds b1 ~ is connected;
:: ORDERS_1:th 189
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_reflexive_in b2
holds b1 ~ is_reflexive_in b2;
:: ORDERS_1:th 190
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_transitive_in b2
holds b1 ~ is_transitive_in b2;
:: ORDERS_1:th 191
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_antisymmetric_in b2
holds b1 ~ is_antisymmetric_in b2;
:: ORDERS_1:th 192
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_connected_in b2
holds b1 ~ is_connected_in b2;
:: ORDERS_1:th 193
theorem
for b1 being Relation-like set
for b2 being set holds
(b1 |_2 b2) ~ = b1 ~ |_2 b2;
:: ORDERS_1:th 194
theorem
for b1 being Relation-like set holds
b1 |_2 {} = {};
:: ORDERS_1:th 195
theorem
for b1 being Relation-like Function-like set
for b2 being set
st b2 is finite & b2 c= proj2 b1
holds ex b3 being set st
b3 c= proj1 b1 & b3 is finite & b1 .: b3 = b2;
:: ORDERS_1:th 196
theorem
for b1 being Relation-like set
st field b1 is finite
holds b1 is finite;
:: ORDERS_1:th 197
theorem
for b1 being Relation-like set
st proj1 b1 is finite & proj2 b1 is finite
holds b1 is finite;