Article WELLORD2, MML version 4.99.1005
:: WELLORD2:funcnot 1 => WELLORD2:func 1
definition
let a1 be set;
func RelIncl A1 -> Relation-like set means
field it = a1 &
(for b1, b2 being set
st b1 in a1 & b2 in a1
holds [b1,b2] in it
iff
b1 c= b2);
end;
:: WELLORD2:def 1
theorem
for b1 being set
for b2 being Relation-like set holds
b2 = RelIncl b1
iff
field b2 = b1 &
(for b3, b4 being set
st b3 in b1 & b4 in b1
holds [b3,b4] in b2
iff
b3 c= b4);
:: WELLORD2:th 2
theorem
for b1 being set holds
RelIncl b1 is reflexive;
:: WELLORD2:th 3
theorem
for b1 being set holds
RelIncl b1 is transitive;
:: WELLORD2:th 4
theorem
for b1 being ordinal set holds
RelIncl b1 is connected;
:: WELLORD2:th 5
theorem
for b1 being set holds
RelIncl b1 is antisymmetric;
:: WELLORD2:th 6
theorem
for b1 being ordinal set holds
RelIncl b1 is well_founded;
:: WELLORD2:th 7
theorem
for b1 being ordinal set holds
RelIncl b1 is well-ordering;
:: WELLORD2:th 8
theorem
for b1, b2 being set
st b1 c= b2
holds (RelIncl b2) |_2 b1 = RelIncl b1;
:: WELLORD2:th 9
theorem
for b1 being ordinal set
for b2 being set
st b2 c= b1
holds RelIncl b2 is well-ordering;
:: WELLORD2:th 10
theorem
for b1, b2 being ordinal set
st b1 in b2
holds b1 = (RelIncl b2) -Seg b1;
:: WELLORD2:th 11
theorem
for b1, b2 being ordinal set
st RelIncl b1,RelIncl b2 are_isomorphic
holds b1 = b2;
:: WELLORD2:th 12
theorem
for b1 being Relation-like set
for b2, b3 being ordinal set
st b1,RelIncl b2 are_isomorphic & b1,RelIncl b3 are_isomorphic
holds b2 = b3;
:: WELLORD2:th 13
theorem
for b1 being Relation-like set
st b1 is well-ordering &
(for b2 being set
st b2 in field b1
holds ex b3 being ordinal set st
b1 |_2 (b1 -Seg b2),RelIncl b3 are_isomorphic)
holds ex b2 being ordinal set st
b1,RelIncl b2 are_isomorphic;
:: WELLORD2:th 14
theorem
for b1 being Relation-like set
st b1 is well-ordering
holds ex b2 being ordinal set st
b1,RelIncl b2 are_isomorphic;
:: WELLORD2:funcnot 2 => WELLORD2:func 2
definition
let a1 be Relation-like set;
assume a1 is well-ordering;
func order_type_of A1 -> ordinal set means
a1,RelIncl it are_isomorphic;
end;
:: WELLORD2:def 2
theorem
for b1 being Relation-like set
st b1 is well-ordering
for b2 being ordinal set holds
b2 = order_type_of b1
iff
b1,RelIncl b2 are_isomorphic;
:: WELLORD2:prednot 1 => WELLORD2:pred 1
definition
let a1 be ordinal set;
let a2 be Relation-like set;
pred A1 is_order_type_of A2 means
a1 = order_type_of a2;
end;
:: WELLORD2:dfs 3
definiens
let a1 be ordinal set;
let a2 be Relation-like set;
To prove
a1 is_order_type_of a2
it is sufficient to prove
thus a1 = order_type_of a2;
:: WELLORD2:def 3
theorem
for b1 being ordinal set
for b2 being Relation-like set holds
b1 is_order_type_of b2
iff
b1 = order_type_of b2;
:: WELLORD2:th 17
theorem
for b1 being set
for b2 being ordinal set
st b1 c= b2
holds order_type_of RelIncl b1 c= b2;
:: WELLORD2:prednot 2 => WELLORD2:pred 2
definition
let a1, a2 be set;
redefine pred A1,A2 are_equipotent means
ex b1 being Relation-like Function-like set st
b1 is one-to-one & proj1 b1 = a1 & proj2 b1 = a2;
symmetry;
:: for a1, a2 being set
:: st a1,a2 are_equipotent
:: holds a2,a1 are_equipotent;
reflexivity;
:: for a1 being set holds
:: a1,a1 are_equipotent;
end;
:: WELLORD2:dfs 4
definiens
let a1, a2 be set;
To prove
a1,a2 are_equipotent
it is sufficient to prove
thus ex b1 being Relation-like Function-like set st
b1 is one-to-one & proj1 b1 = a1 & proj2 b1 = a2;
:: WELLORD2:def 4
theorem
for b1, b2 being set holds
b1,b2 are_equipotent
iff
ex b3 being Relation-like Function-like set st
b3 is one-to-one & proj1 b3 = b1 & proj2 b3 = b2;
:: WELLORD2:th 22
theorem
for b1, b2, b3 being set
st b1,b2 are_equipotent & b2,b3 are_equipotent
holds b1,b3 are_equipotent;
:: WELLORD2:th 25
theorem
for b1 being set
for b2 being Relation-like set
st b2 well_orders b1
holds field (b2 |_2 b1) = b1 & b2 |_2 b1 is well-ordering;
:: WELLORD2:th 26
theorem
for b1 being set holds
ex b2 being Relation-like set st
b2 well_orders b1;
:: WELLORD2:th 27
theorem
for b1 being non empty set
st (for b2 being set
st b2 in b1
holds b2 <> {}) &
(for b2, b3 being set
st b2 in b1 & b3 in b1 & b2 <> b3
holds b2 misses b3)
holds ex b2 being set st
for b3 being set
st b3 in b1
holds ex b4 being set st
b2 /\ b3 = {b4};
:: WELLORD2:th 28
theorem
for b1 being non empty set
st for b2 being set
st b2 in b1
holds b2 <> {}
holds ex b2 being Relation-like Function-like set st
proj1 b2 = b1 &
(for b3 being set
st b3 in b1
holds b2 . b3 in b3);
:: WELLORD2:sch 1
scheme WELLORD2:sch 1
{F1 -> set,
F2 -> set}:
ex b1 being Relation-like Function-like set st
proj1 b1 = F1() &
proj2 b1 c= F2() &
(for b2 being set
st b2 in F1()
holds P1[b2, b1 . b2])
provided
for b1 being set
st b1 in F1()
holds ex b2 being set st
b2 in F2() & P1[b1, b2];
:: WELLORD2:sch 2
scheme WELLORD2:sch 2
{F1 -> set,
F2 -> set,
F3 -> set}:
ex b1, b2 being Relation-like Function-like set st
proj1 b1 = F1() &
proj1 b2 = F1() &
(for b3 being set
st b3 in F1()
holds P1[b3, b1 . b3, b2 . b3])
provided
for b1 being set
st b1 in F1()
holds ex b2, b3 being set st
b2 in F2() & b3 in F3() & P1[b1, b2, b3];
:: WELLORD2:th 29
theorem
for b1 being set holds
RelIncl b1 is_reflexive_in b1;
:: WELLORD2:th 30
theorem
for b1 being set holds
RelIncl b1 is_transitive_in b1;
:: WELLORD2:th 31
theorem
for b1 being set holds
RelIncl b1 is_antisymmetric_in b1;