:: The Ordinal Numbers. Transfinite Induction and Defining by Transfinite Induction
:: by Grzegorz Bancerek
::
:: Received March 20, 1989
:: Copyright (c) 1990 Association of Mizar Users

begin

theorem :: ORDINAL1:1

canceled;

theorem :: ORDINAL1:2

canceled;

theorem Th3: :: ORDINAL1:3

for X, Y, Z being set holds
( not X in Y or not Y in Z or not Z in X )

proof end;

theorem :: ORDINAL1:4

for X1, X2, X3, X4 being set holds
( not X1 in X2 or not X2 in X3 or not X3 in X4 or not X4 in X1 )

proof end;

theorem :: ORDINAL1:5

for X1, X2, X3, X4, X5 being set holds
( not X1 in X2 or not X2 in X3 or not X3 in X4 or not X4 in X5 or not X5 in X1 )

proof end;

theorem :: ORDINAL1:6

for X1, X2, X3, X4, X5, X6 being set holds
( not X1 in X2 or not X2 in X3 or not X3 in X4 or not X4 in X5 or not X5 in X6 or not X6 in X1 )

proof end;

theorem Th7: :: ORDINAL1:7

for Y, X being set st Y in X holds
not X c= Y

proof end;

definition

let X be set ;
func succ X -> set equals :: ORDINAL1:def 1
X \/ {X};
coherence
X \/ {X} is set ;

end;

:: deftheorem defines succ ORDINAL1:def 1 :
for X being set holds succ X = X \/ {X};

registration

let X be set ;
cluster succ X -> non empty ;
coherence
not succ X is empty ;

end;

theorem :: ORDINAL1:8

canceled;

theorem :: ORDINAL1:9

canceled;

theorem Th10: :: ORDINAL1:10

for X being set holds X in succ X

proof end;

theorem :: ORDINAL1:11

canceled;

theorem :: ORDINAL1:12

for X, Y being set st succ X = succ Y holds
X = Y

proof end;

theorem Th13: :: ORDINAL1:13

for x, X being set holds
( x in succ X iff ( x in X or x = X ) )

proof end;

theorem Th14: :: ORDINAL1:14

for X being set holds X <> succ X

proof end;

definition

let X be set ;
attr X is epsilon-transitive means :Def2: :: ORDINAL1:def 2
for x being set st x in X holds
x c= X;
attr X is epsilon-connected means :Def3: :: ORDINAL1:def 3
for x, y being set st x in X & y in X & not x in y & not x = y holds
y in x;

end;

:: deftheorem Def2 defines epsilon-transitive ORDINAL1:def 2 :
for X being set holds
( X is epsilon-transitive iff for x being set st x in X holds
x c= X );

:: deftheorem Def3 defines epsilon-connected ORDINAL1:def 3 :
for X being set holds
( X is epsilon-connected iff for x, y being set st x in X & y in X & not x in y & not x = y holds
y in x );

Lm1: ( {} is epsilon-transitive & {} is epsilon-connected )

proof end;

definition

let IT be set ;
attr IT is ordinal means :Def4: :: ORDINAL1:def 4
( IT is epsilon-transitive & IT is epsilon-connected );

end;

:: deftheorem Def4 defines ordinal ORDINAL1:def 4 :
for IT being set holds
( IT is ordinal iff ( IT is epsilon-transitive & IT is epsilon-connected ) );

registration

cluster ordinal -> epsilon-transitive epsilon-connected number ;
coherence
for b₁ being set st b₁ is ordinal holds
( b₁ is epsilon-transitive & b₁ is epsilon-connected ) by Def4;
cluster epsilon-transitive epsilon-connected -> ordinal number ;
coherence
for b₁ being set st b₁ is epsilon-transitive & b₁ is epsilon-connected holds
b₁ is ordinal by Def4;

end;

notation

synonym number for set ;

end;

registration

cluster ordinal number ;
existence
ex b₁ being number st b₁ is ordinal by Lm1;

end;

definition

mode Ordinal is ordinal number ;

end;

theorem :: ORDINAL1:15

canceled;

theorem :: ORDINAL1:16

canceled;

theorem :: ORDINAL1:17

canceled;

theorem :: ORDINAL1:18

canceled;

theorem Th19: :: ORDINAL1:19

for A, B being set
for C being epsilon-transitive set st A in B & B in C holds
A in C

proof end;

theorem :: ORDINAL1:20

canceled;

theorem Th21: :: ORDINAL1:21

for x being epsilon-transitive set
for A being Ordinal st x c< A holds
x in A

proof end;

theorem :: ORDINAL1:22

for A being epsilon-transitive set
for B, C being Ordinal st A c= B & B in C holds
A in C

proof end;

theorem Th23: :: ORDINAL1:23

for a being set
for A being Ordinal st a in A holds
a is Ordinal

proof end;

theorem Th24: :: ORDINAL1:24

for A, B being Ordinal holds
( A in B or A = B or B in A )

proof end;

definition

let A, B be Ordinal;
:: original: c=
redefine pred A c= B means :: ORDINAL1:def 5
for C being Ordinal st C in A holds
C in B;
compatibility
( A c= B iff for C being Ordinal st C in A holds
C in B )

proof end;

connectedness
for A, B being Ordinal st ex C being Ordinal st
( C in A & not C in B ) holds
for C being Ordinal st C in B holds
C in A

proof end;

end;

:: deftheorem defines c= ORDINAL1:def 5 :
for A, B being Ordinal holds
( A c= B iff for C being Ordinal st C in A holds
C in B );

theorem :: ORDINAL1:25

for A, B being Ordinal holds A,B are_c=-comparable

proof end;

theorem Th26: :: ORDINAL1:26

for A, B being Ordinal holds
( A c= B or B in A )

proof end;

registration

cluster empty -> ordinal number ;
coherence
for b₁ being number st b₁ is empty holds
b₁ is ordinal by Lm1;

end;

theorem :: ORDINAL1:27

canceled;

theorem :: ORDINAL1:28

canceled;

theorem Th29: :: ORDINAL1:29

for x being set st x is Ordinal holds
succ x is Ordinal

proof end;

theorem Th30: :: ORDINAL1:30

for x being set st x is ordinal holds
union x is ordinal

proof end;

registration

cluster non empty epsilon-transitive epsilon-connected ordinal number ;
existence
not for b₁ being Ordinal holds b₁ is empty

proof end;

end;

registration

let A be Ordinal;
cluster succ A -> non empty ordinal ;
coherence
( not succ A is empty & succ A is ordinal ) by Th29;
cluster union A -> ordinal ;
coherence
union A is ordinal by Th30;

end;

theorem Th31: :: ORDINAL1:31

for X being set st ( for x being set st x in X holds
( x is Ordinal & x c= X ) ) holds
X is ordinal

proof end;

theorem Th32: :: ORDINAL1:32

for X being set
for A being Ordinal st X c= A & X <> {} holds
ex C being Ordinal st
( C in X & ( for B being Ordinal st B in X holds
C c= B ) )

proof end;

theorem Th33: :: ORDINAL1:33

for A, B being Ordinal holds
( A in B iff succ A c= B )

proof end;

theorem Th34: :: ORDINAL1:34

for A, C being Ordinal holds
( A in succ C iff A c= C )

proof end;

scheme :: ORDINAL1:sch 1
OrdinalMin{ P₁[ Ordinal] } :

ex A being Ordinal st
( P₁[A] & ( for B being Ordinal st P₁[B] holds
A c= B ) )

provided

A1: ex A being Ordinal st P₁[A]

proof end;

scheme :: ORDINAL1:sch 2
TransfiniteInd{ P₁[ Ordinal] } :

for A being Ordinal holds P₁[A]

provided

A1: for A being Ordinal st ( for C being Ordinal st C in A holds
P₁[C] ) holds
P₁[A]

proof end;

theorem Th35: :: ORDINAL1:35

for X being set st ( for a being set st a in X holds
a is Ordinal ) holds
union X is ordinal

proof end;

theorem Th36: :: ORDINAL1:36

for X being set st ( for a being set st a in X holds
a is Ordinal ) holds
ex A being Ordinal st X c= A

proof end;

theorem Th37: :: ORDINAL1:37

for X being set holds
not for x being set holds
( x in X iff x is Ordinal )

proof end;

theorem Th38: :: ORDINAL1:38

for X being set holds
not for A being Ordinal holds A in X

proof end;

theorem :: ORDINAL1:39

for X being set ex A being Ordinal st
( not A in X & ( for B being Ordinal st not B in X holds
A c= B ) )

proof end;

definition

let A be set ;
attr A is limit_ordinal means :Def6: :: ORDINAL1:def 6
A = union A;

end;

:: deftheorem Def6 defines limit_ordinal ORDINAL1:def 6 :
for A being set holds
( A is limit_ordinal iff A = union A );

theorem :: ORDINAL1:40

canceled;

theorem Th41: :: ORDINAL1:41

for A being Ordinal holds
( A is limit_ordinal iff for C being Ordinal st C in A holds
succ C in A )

proof end;

theorem :: ORDINAL1:42

for A being Ordinal holds
( not A is limit_ordinal iff ex B being Ordinal st A = succ B )

proof end;

definition

let IT be set ;
attr IT is T-Sequence-like means :Def7: :: ORDINAL1:def 7
proj1 IT is ordinal ;

end;

:: deftheorem Def7 defines T-Sequence-like ORDINAL1:def 7 :
for IT being set holds
( IT is T-Sequence-like iff proj1 IT is ordinal );

registration

cluster empty -> T-Sequence-like number ;
coherence
for b₁ being set st b₁ is empty holds
b₁ is T-Sequence-like

proof end;

end;

definition

mode T-Sequence is T-Sequence-like Function;

end;

registration

let Z be set ;
cluster Relation-like Z -valued Function-like T-Sequence-like number ;
existence
ex b₁ being T-Sequence st b₁ is Z -valued

proof end;

end;

definition

let Z be set ;
mode T-Sequence of Z is Z -valued T-Sequence;

end;

theorem :: ORDINAL1:43

canceled;

theorem :: ORDINAL1:44

canceled;

theorem :: ORDINAL1:45

for Z being set holds {} is T-Sequence of Z

proof end;

theorem :: ORDINAL1:46

for F being Function st dom F is Ordinal holds
F is T-Sequence of rng F by Def7, RELAT_1:def 19;

registration

let L be T-Sequence;
cluster proj1 L -> ordinal ;
coherence
dom L is ordinal by Def7;

end;

theorem :: ORDINAL1:47

for X, Y being set st X c= Y holds
for L being T-Sequence of X holds L is T-Sequence of Y

proof end;

registration

let L be T-Sequence;
let A be Ordinal;
cluster L | A -> rng L -valued T-Sequence-like ;
coherence
( L | A is rng L -valued & L | A is T-Sequence-like )

proof end;

end;

theorem :: ORDINAL1:48

for X being set
for L being T-Sequence of X
for A being Ordinal holds L | A is T-Sequence of X ;

definition

canceled;
let IT be set ;
attr IT is c=-linear means :Def9: :: ORDINAL1:def 9
for x, y being set st x in IT & y in IT holds
x,y are_c=-comparable ;

end;

:: deftheorem ORDINAL1:def 8 :
canceled;

:: deftheorem Def9 defines c=-linear ORDINAL1:def 9 :
for IT being set holds
( IT is c=-linear iff for x, y being set st x in IT & y in IT holds
x,y are_c=-comparable );

theorem :: ORDINAL1:49

for X being set st ( for a being set st a in X holds
a is T-Sequence ) & X is c=-linear holds
union X is T-Sequence

proof end;

scheme :: ORDINAL1:sch 3
TSUniq{ F₁() -> Ordinal, F₂( T-Sequence) -> set , F₃() -> T-Sequence, F₄() -> T-Sequence } :

F₃() = F₄()

provided

A1: ( dom F₃() = F₁() & ( for B being Ordinal
for L being T-Sequence st B in F₁() & L = F₃() | B holds
F₃() . B = F₂(L) ) ) and
A2: ( dom F₄() = F₁() & ( for B being Ordinal
for L being T-Sequence st B in F₁() & L = F₄() | B holds
F₄() . B = F₂(L) ) )

proof end;

scheme :: ORDINAL1:sch 4
TSExist{ F₁() -> Ordinal, F₂( T-Sequence) -> set } :

ex L being T-Sequence st
( dom L = F₁() & ( for B being Ordinal
for L1 being T-Sequence st B in F₁() & L1 = L | B holds
L . B = F₂(L1) ) )

proof end;

scheme :: ORDINAL1:sch 5
FuncTS{ F₁() -> T-Sequence, F₂( Ordinal) -> set , F₃( T-Sequence) -> set } :

for B being Ordinal st B in dom F₁() holds
F₁() . B = F₃((F₁() | B))

provided

A1: for A being Ordinal
for a being set holds
( a = F₂(A) iff ex L being T-Sequence st
( a = F₃(L) & dom L = A & ( for B being Ordinal st B in A holds
L . B = F₃((L | B)) ) ) ) and
A2: for A being Ordinal st A in dom F₁() holds
F₁() . A = F₂(A)

proof end;

theorem :: ORDINAL1:50

for A, B being Ordinal holds
( A c< B or A = B or B c< A )

proof end;

begin

definition

let X be set ;
func On X -> set means :Def10: :: ORDINAL1:def 10
for x being set holds
( x in it iff ( x in X & x is Ordinal ) );
existence
ex b₁ being set st
for x being set holds
( x in b₁ iff ( x in X & x is Ordinal ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff ( x in X & x is Ordinal ) ) ) & ( for x being set holds
( x in b₂ iff ( x in X & x is Ordinal ) ) ) holds
b₁ = b₂

proof end;

func Lim X -> set means :: ORDINAL1:def 11
for x being set holds
( x in it iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) );
existence
ex b₁ being set st
for x being set holds
( x in b₁ iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) ) ) & ( for x being set holds
( x in b₂ iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines On ORDINAL1:def 10 :
for X, b₂ being set holds
( b₂ = On X iff for x being set holds
( x in b₂ iff ( x in X & x is Ordinal ) ) );

:: deftheorem defines Lim ORDINAL1:def 11 :
for X, b₂ being set holds
( b₂ = Lim X iff for x being set holds
( x in b₂ iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) ) );

theorem Th51: :: ORDINAL1:51

for D being Ordinal ex A being Ordinal st
( D in A & A is limit_ordinal )

proof end;

definition

func omega -> set means :Def12: :: ORDINAL1:def 12
( {} in it & it is limit_ordinal & it is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
it c= A ) );
existence
ex b₁ being set st
( {} in b₁ & b₁ is limit_ordinal & b₁ is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b₁ c= A ) )

proof end;

uniqueness
for b₁, b₂ being set st {} in b₁ & b₁ is limit_ordinal & b₁ is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b₁ c= A ) & {} in b₂ & b₂ is limit_ordinal & b₂ is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b₂ c= A ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines omega ORDINAL1:def 12 :
for b₁ being set holds
( b₁ = omega iff ( {} in b₁ & b₁ is limit_ordinal & b₁ is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
b₁ c= A ) ) );

registration

cluster omega -> non empty ordinal ;
coherence
( not omega is empty & omega is ordinal ) by Def12;

end;

definition

let A be set ;
attr A is natural means :Def13: :: ORDINAL1:def 13
A in omega ;

end;

:: deftheorem Def13 defines natural ORDINAL1:def 13 :
for A being set holds
( A is natural iff A in omega );

registration

cluster natural number ;
existence
ex b₁ being number st b₁ is natural

proof end;

end;

registration

let A be Ordinal;
cluster -> ordinal Element of A;
coherence
for b₁ being Element of A holds b₁ is ordinal

proof end;

end;

registration

cluster natural -> ordinal number ;
coherence
for b₁ being number st b₁ is natural holds
b₁ is ordinal

proof end;

end;

scheme :: ORDINAL1:sch 6
ALFA{ F₁() -> non empty set , P₁[ set , set ] } :

ex F being Function st
( dom F = F₁() & ( for d being Element of F₁() ex A being Ordinal st
( A = F . d & P₁[d,A] & ( for B being Ordinal st P₁[d,B] holds
A c= B ) ) ) )

provided

A1: for d being Element of F₁() ex A being Ordinal st P₁[d,A]

proof end;

theorem :: ORDINAL1:52

for X being set holds (succ X) \ {X} = X

proof end;

registration

cluster empty -> natural number ;
coherence
for b₁ being number st b₁ is empty holds
b₁ is natural

proof end;

cluster -> natural Element of omega ;
coherence
for b₁ being Element of omega holds b₁ is natural by Def13;

end;

registration

cluster non empty natural number ;
existence
ex b₁ being number st
( not b₁ is empty & b₁ is natural )

proof end;

end;

registration

let a be natural Ordinal;
cluster succ a -> natural ;
coherence
succ a is natural

proof end;

end;