Article CARD_LAR, MML version 4.99.1005
:: CARD_LAR:condreg 1
registration
cluster ordinal infinite cardinal -> being_limit_ordinal (set);
end;
:: CARD_LAR:condreg 2
registration
cluster ordinal being_limit_ordinal non empty -> infinite (set);
end;
:: CARD_LAR:condreg 3
registration
cluster infinite cardinal non limit -> non countable (set);
end;
:: CARD_LAR:exreg 1
registration
cluster epsilon-transitive epsilon-connected ordinal being_limit_ordinal non empty infinite cardinal non countable regular set;
end;
:: CARD_LAR:prednot 1 => CARD_LAR:pred 1
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
pred A2 is_unbounded_in A1 means
a2 c= a1 & sup a2 = a1;
end;
:: CARD_LAR:dfs 1
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
To prove
a2 is_unbounded_in a1
it is sufficient to prove
thus a2 c= a1 & sup a2 = a1;
:: CARD_LAR:def 1
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being set holds
b2 is_unbounded_in b1
iff
b2 c= b1 & sup b2 = b1;
:: CARD_LAR:prednot 2 => CARD_LAR:pred 2
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
pred A2 is_closed_in A1 means
a2 c= a1 &
(for b1 being ordinal being_limit_ordinal infinite set
st b1 in a1 & sup (a2 /\ b1) = b1
holds b1 in a2);
end;
:: CARD_LAR:dfs 2
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
To prove
a2 is_closed_in a1
it is sufficient to prove
thus a2 c= a1 &
(for b1 being ordinal being_limit_ordinal infinite set
st b1 in a1 & sup (a2 /\ b1) = b1
holds b1 in a2);
:: CARD_LAR:def 2
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being set holds
b2 is_closed_in b1
iff
b2 c= b1 &
(for b3 being ordinal being_limit_ordinal infinite set
st b3 in b1 & sup (b2 /\ b3) = b3
holds b3 in b2);
:: CARD_LAR:prednot 3 => CARD_LAR:pred 3
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
pred A2 is_club_in A1 means
a2 is_closed_in a1 & a2 is_unbounded_in a1;
end;
:: CARD_LAR:dfs 3
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
To prove
a2 is_club_in a1
it is sufficient to prove
thus a2 is_closed_in a1 & a2 is_unbounded_in a1;
:: CARD_LAR:def 3
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being set holds
b2 is_club_in b1
iff
b2 is_closed_in b1 & b2 is_unbounded_in b1;
:: CARD_LAR:attrnot 1 => CARD_LAR:attr 1
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
attr a2 is unbounded means
sup a2 = a1;
end;
:: CARD_LAR:dfs 4
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
To prove
a2 is unbounded
it is sufficient to prove
thus sup a2 = a1;
:: CARD_LAR:def 4
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 is unbounded(b1)
iff
sup b2 = b1;
:: CARD_LAR:attrnot 2 => CARD_LAR:attr 2
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
attr a2 is closed means
for b1 being ordinal being_limit_ordinal infinite set
st b1 in a1 & sup (a2 /\ b1) = b1
holds b1 in a2;
end;
:: CARD_LAR:dfs 5
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
To prove
a2 is closed
it is sufficient to prove
thus for b1 being ordinal being_limit_ordinal infinite set
st b1 in a1 & sup (a2 /\ b1) = b1
holds b1 in a2;
:: CARD_LAR:def 5
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 is closed(b1)
iff
for b3 being ordinal being_limit_ordinal infinite set
st b3 in b1 & sup (b2 /\ b3) = b3
holds b3 in b2;
:: CARD_LAR:attrnot 3 => CARD_LAR:attr 1
notation
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
antonym bounded for unbounded;
end;
:: CARD_LAR:th 2
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 is_club_in b1
iff
b2 is closed(b1) & b2 is unbounded(b1);
:: CARD_LAR:th 3
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 c= sup b2;
:: CARD_LAR:th 4
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1
st b2 is not empty &
(for b3 being ordinal set
st b3 in b2
holds ex b4 being ordinal set st
b4 in b2 & b3 in b4)
holds sup b2 is ordinal being_limit_ordinal infinite set;
:: CARD_LAR:th 5
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 is bounded(b1)
iff
ex b3 being ordinal set st
b3 in b1 & b2 c= b3;
:: CARD_LAR:th 6
theorem
for b1, b2 being ordinal being_limit_ordinal infinite set
for b3 being Element of bool b1
st sup (b3 /\ b2) <> b2
holds ex b4 being ordinal set st
b4 in b2 & b3 /\ b2 c= b4;
:: CARD_LAR:th 7
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 is unbounded(b1)
iff
for b3 being ordinal set
st b3 in b1
holds ex b4 being ordinal set st
b4 in b2 & b3 c= b4;
:: CARD_LAR:th 8
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1
st b2 is unbounded(b1)
holds b2 is not empty;
:: CARD_LAR:th 9
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being ordinal set
for b3 being Element of bool b1
st b3 is unbounded(b1) & b2 in b1
holds ex b4 being Element of b1 st
b4 in {b5 where b5 is Element of b1: b5 in b3 & b2 in b5};
:: CARD_LAR:funcnot 1 => CARD_LAR:func 1
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
let a3 be ordinal set;
assume a2 is unbounded(a1) & a3 in a1;
func LBound(A3,A2) -> Element of a2 equals
inf {b1 where b1 is Element of a1: b1 in a2 & a3 in b1};
end;
:: CARD_LAR:def 6
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1
for b3 being ordinal set
st b2 is unbounded(b1) & b3 in b1
holds LBound(b3,b2) = inf {b4 where b4 is Element of b1: b4 in b2 & b3 in b4};
:: CARD_LAR:th 10
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being ordinal set
for b3 being Element of bool b1
st b3 is unbounded(b1) & b2 in b1
holds LBound(b2,b3) in b3 & b2 in LBound(b2,b3);
:: CARD_LAR:th 11
theorem
for b1 being ordinal being_limit_ordinal infinite set holds
[#] b1 is closed(b1) & [#] b1 is unbounded(b1);
:: CARD_LAR:th 12
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being ordinal set
for b3 being Element of bool b1
st b2 in b1 & b3 is closed(b1) & b3 is unbounded(b1)
holds b3 \ b2 is closed(b1) & b3 \ b2 is unbounded(b1);
:: CARD_LAR:th 13
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being ordinal set
st b2 in b1
holds b1 \ b2 is closed(b1) & b1 \ b2 is unbounded(b1);
:: CARD_LAR:attrnot 4 => CARD_LAR:attr 3
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
attr a2 is stationary means
for b1 being Element of bool a1
st b1 is closed(a1) & b1 is unbounded(a1)
holds a2 /\ b1 is not empty;
end;
:: CARD_LAR:dfs 7
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
To prove
a2 is stationary
it is sufficient to prove
thus for b1 being Element of bool a1
st b1 is closed(a1) & b1 is unbounded(a1)
holds a2 /\ b1 is not empty;
:: CARD_LAR:def 7
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
b2 is stationary(b1)
iff
for b3 being Element of bool b1
st b3 is closed(b1) & b3 is unbounded(b1)
holds b2 /\ b3 is not empty;
:: CARD_LAR:th 14
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2, b3 being Element of bool b1
st b2 is stationary(b1) & b2 c= b3
holds b3 is stationary(b1);
:: CARD_LAR:prednot 4 => CARD_LAR:pred 4
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
pred A2 is_stationary_in A1 means
a2 c= a1 &
(for b1 being Element of bool a1
st b1 is closed(a1) & b1 is unbounded(a1)
holds a2 /\ b1 is not empty);
end;
:: CARD_LAR:dfs 8
definiens
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be set;
To prove
a2 is_stationary_in a1
it is sufficient to prove
thus a2 c= a1 &
(for b1 being Element of bool a1
st b1 is closed(a1) & b1 is unbounded(a1)
holds a2 /\ b1 is not empty);
:: CARD_LAR:def 8
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being set holds
b2 is_stationary_in b1
iff
b2 c= b1 &
(for b3 being Element of bool b1
st b3 is closed(b1) & b3 is unbounded(b1)
holds b2 /\ b3 is not empty);
:: CARD_LAR:th 15
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2, b3 being set
st b2 is_stationary_in b1 & b2 c= b3 & b3 c= b1
holds b3 is_stationary_in b1;
:: CARD_LAR:modenot 1 => CARD_LAR:mode 1
definition
let a1 be set;
let a2 be Element of bool bool a1;
redefine mode Element of a2 -> Element of bool a1;
end;
:: CARD_LAR:th 16
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1
st b2 is stationary(b1)
holds b2 is unbounded(b1);
:: CARD_LAR:funcnot 2 => CARD_LAR:func 2
definition
let a1 be ordinal being_limit_ordinal infinite set;
let a2 be Element of bool a1;
func limpoints A2 -> Element of bool a1 equals
{b1 where b1 is Element of a1: b1 is infinite & b1 is being_limit_ordinal & sup (a2 /\ b1) = b1};
end;
:: CARD_LAR:def 9
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
limpoints b2 = {b3 where b3 is Element of b1: b3 is infinite & b3 is being_limit_ordinal & sup (b2 /\ b3) = b3};
:: CARD_LAR:th 17
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2, b3 being ordinal set
for b4 being Element of bool b1
st b4 /\ b2 c= b3
holds b2 /\ limpoints b4 c= succ b3;
:: CARD_LAR:th 18
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being ordinal set
for b3 being Element of bool b1
st b3 c= b2
holds limpoints b3 c= succ b2;
:: CARD_LAR:th 19
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1 holds
limpoints b2 is closed(b1);
:: CARD_LAR:th 20
theorem
for b1 being ordinal being_limit_ordinal infinite set
for b2 being Element of bool b1
st b2 is unbounded(b1) & limpoints b2 is bounded(b1)
holds ex b3 being ordinal set st
b3 in b1 &
{succ b4 where b4 is Element of b1: b4 in b2 & b3 in succ b4} is_club_in b1;
:: CARD_LAR:exreg 2
registration
let a1 be infinite cardinal non countable set;
cluster epsilon-transitive epsilon-connected ordinal infinite cardinal Element of a1;
end;
:: CARD_LAR:th 21
theorem
for b1 being infinite cardinal set
for b2 being Element of bool b1
st b2 is unbounded(b1)
holds cf b1 c= Card b2;
:: CARD_LAR:th 22
theorem
for b1 being infinite cardinal non countable set
for b2 being Element of bool bool b1
st for b3 being Element of b2 holds
b3 is closed(b1)
holds meet b2 is closed(b1);
:: CARD_LAR:th 23
theorem
for b1 being infinite cardinal non countable set
for b2 being Element of bool b1
st alef 0 in cf b1
for b3 being Function-like quasi_total Relation of NAT,b2 holds
sup proj2 b3 in b1;
:: CARD_LAR:th 24
theorem
for b1 being infinite cardinal non countable set
st alef 0 in cf b1
for b2 being non empty Element of bool bool b1
st Card b2 in cf b1 &
(for b3 being Element of b2 holds
b3 is closed(b1) & b3 is unbounded(b1))
holds meet b2 is closed(b1) & meet b2 is unbounded(b1);
:: CARD_LAR:th 25
theorem
for b1 being infinite cardinal non countable set
for b2 being Element of bool b1
st alef 0 in cf b1 & b2 is unbounded(b1)
for b3 being ordinal set
st b3 in b1
holds ex b4 being ordinal being_limit_ordinal infinite set st
b4 in b1 & b3 in b4 & b4 in limpoints b2;
:: CARD_LAR:th 26
theorem
for b1 being infinite cardinal non countable set
for b2 being Element of bool b1
st alef 0 in cf b1 & b2 is unbounded(b1)
holds limpoints b2 is unbounded(b1);
:: CARD_LAR:attrnot 5 => CARD_LAR:attr 4
definition
let a1 be infinite cardinal non countable set;
attr a1 is Mahlo means
{b1 where b1 is infinite cardinal Element of a1: b1 is regular} is_stationary_in a1;
end;
:: CARD_LAR:dfs 10
definiens
let a1 be infinite cardinal non countable set;
To prove
a1 is Mahlo
it is sufficient to prove
thus {b1 where b1 is infinite cardinal Element of a1: b1 is regular} is_stationary_in a1;
:: CARD_LAR:def 10
theorem
for b1 being infinite cardinal non countable set holds
b1 is Mahlo
iff
{b2 where b2 is infinite cardinal Element of b1: b2 is regular} is_stationary_in b1;
:: CARD_LAR:attrnot 6 => CARD_LAR:attr 5
definition
let a1 be infinite cardinal non countable set;
attr a1 is strongly_Mahlo means
{b1 where b1 is infinite cardinal Element of a1: b1 is strongly_inaccessible} is_stationary_in a1;
end;
:: CARD_LAR:dfs 11
definiens
let a1 be infinite cardinal non countable set;
To prove
a1 is strongly_Mahlo
it is sufficient to prove
thus {b1 where b1 is infinite cardinal Element of a1: b1 is strongly_inaccessible} is_stationary_in a1;
:: CARD_LAR:def 11
theorem
for b1 being infinite cardinal non countable set holds
b1 is strongly_Mahlo
iff
{b2 where b2 is infinite cardinal Element of b1: b2 is strongly_inaccessible} is_stationary_in b1;
:: CARD_LAR:prednot 5 => CARD_LAR:attr 4
notation
let a1 be infinite cardinal non countable set;
synonym a1 is_Mahlo for Mahlo;
end;
:: CARD_LAR:prednot 6 => CARD_LAR:attr 5
notation
let a1 be infinite cardinal non countable set;
synonym a1 is_strongly_Mahlo for strongly_Mahlo;
end;
:: CARD_LAR:th 27
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_Mahlo
holds b1 is Mahlo;
:: CARD_LAR:th 28
theorem
for b1 being infinite cardinal non countable set
st b1 is Mahlo
holds b1 is regular;
:: CARD_LAR:th 29
theorem
for b1 being infinite cardinal non countable set
st b1 is Mahlo
holds b1 is limit;
:: CARD_LAR:th 30
theorem
for b1 being infinite cardinal non countable set
st b1 is Mahlo
holds b1 is inaccessible;
:: CARD_LAR:th 31
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_Mahlo
holds b1 is strong_limit;
:: CARD_LAR:th 32
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_Mahlo
holds b1 is strongly_inaccessible;
:: CARD_LAR:th 33
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds ex b3 being set st
b3 in b1 & b2 c= b3 & b3 is cardinal set
holds union b1 is cardinal set;
:: CARD_LAR:th 34
theorem
for b1 being set
for b2 being infinite cardinal set
st Card b1 in cf b2 &
(for b3 being set
st b3 in b1
holds Card b3 in b2)
holds Card union b1 in b2;
:: CARD_LAR:th 35
theorem
for b1 being infinite cardinal non countable set
for b2 being ordinal set
st b1 is strongly_inaccessible & b2 in b1
holds Card Rank b2 in b1;
:: CARD_LAR:th 36
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_inaccessible
holds Card Rank b1 = b1;
:: CARD_LAR:th 37
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_inaccessible
holds Rank b1 is being_Tarski-Class;
:: CARD_LAR:th 38
theorem
for b1 being ordinal non empty set holds
Rank b1 is not empty;
:: CARD_LAR:funcreg 1
registration
let a1 be ordinal non empty set;
cluster Rank a1 -> non empty;
end;
:: CARD_LAR:th 39
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_inaccessible
holds Rank b1 is non empty universal set;
:: CARD_LAR:th 40
theorem
for b1 being infinite cardinal non countable set
st b1 is strongly_inaccessible
holds Rank b1 is being_a_model_of_ZF;