HomeHome Metamath Proof Explorer
Theorem List (p. 75 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoemapval 7401* Value of the relation  T. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   =>    |-  ( ph  ->  ( F T G  <->  E. z  e.  B  ( ( F `  z )  e.  ( G `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) ) )
 
Theoremoemapvali 7402* If  F  <  G, then there is some  z witnessing this, but we can say more and in fact there is a definable expression  X that also witnesses  F  <  G. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   =>    |-  ( ph  ->  ( X  e.  B  /\  ( F `  X )  e.  ( G `  X )  /\  A. w  e.  B  ( X  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) )
 
Theoremcantnflem1a 7403* Lemma for cantnf 7411. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   =>    |-  ( ph  ->  X  e.  ( `' G " ( _V  \  1o ) ) )
 
Theoremcantnflem1b 7404* Lemma for cantnf 7411. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   =>    |-  ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u ) )  ->  X  C_  ( O `  u ) )
 
Theoremcantnflem1c 7405* Lemma for cantnf 7411. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   =>    |-  ( ( ( (
 ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
 C_  u ) ) 
 /\  x  e.  B )  /\  ( ( F `
  x )  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  ( `' G "
 ( _V  \  1o ) ) )
 
Theoremcantnflem1d 7406* Lemma for cantnf 7411. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( O `
  k ) )  .o  ( G `  ( O `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  ( x  e.  B  |->  if ( x  C_  X ,  ( F `  x ) ,  (/) ) ) )  e.  ( H `
  suc  ( `' O `  X ) ) )
 
Theoremcantnflem1 7407* Lemma for cantnf 7411. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation  T is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct  F ,  G are  T -related as  F  <  G or  G  <  F, and WLOG assuming that  F  <  G, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( O `
  k ) )  .o  ( G `  ( O `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  F )  e.  ( ( A CNF  B ) `  G ) )
 
Theoremcantnflem2 7408* Lemma for cantnf 7411. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  C  e.  ( A  ^o  B ) )   &    |-  ( ph  ->  C  C_  ran  ( A CNF  B ) )   &    |-  ( ph  ->  (/)  e.  C )   =>    |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
 
Theoremcantnflem3 7409* Lemma for cantnf 7411. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than  C has a normal form, we can use oeeu 6617 to factor  C into the form  ( ( A  ^o  X )  .o  Y )  +o  Z where  0  <  Y  <  A and  Z  <  ( A  ^o  X ) (and a fortiori  X  < 
B). Then since  Z  <  ( A  ^o  X )  <_ 
( A  ^o  X
)  .o  Y  <_  C,  Z has a normal form, and by appending the term  ( A  ^o  X )  .o  Y using cantnfp1 7399 we get a normal form for 
C. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  C  e.  ( A  ^o  B ) )   &    |-  ( ph  ->  C  C_  ran  ( A CNF  B ) )   &    |-  ( ph  ->  (/)  e.  C )   &    |-  X  =  U. |^| { c  e.  On  |  C  e.  ( A  ^o  c ) }   &    |-  P  =  (
 iota d E. a  e.  On  E. b  e.  ( A  ^o  X ) ( d  = 
 <. a ,  b >.  /\  ( ( ( A 
 ^o  X )  .o  a )  +o  b
 )  =  C ) )   &    |-  Y  =  ( 1st `  P )   &    |-  Z  =  ( 2nd `  P )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  ( ( A CNF  B ) `
  G )  =  Z )   &    |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `
  t ) ) )   =>    |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
 
Theoremcantnflem4 7410* Lemma for cantnf 7411. Complete the induction step of cantnflem3 7409. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  C  e.  ( A  ^o  B ) )   &    |-  ( ph  ->  C  C_  ran  ( A CNF  B ) )   &    |-  ( ph  ->  (/)  e.  C )   &    |-  X  =  U. |^| { c  e.  On  |  C  e.  ( A  ^o  c ) }   &    |-  P  =  (
 iota d E. a  e.  On  E. b  e.  ( A  ^o  X ) ( d  = 
 <. a ,  b >.  /\  ( ( ( A 
 ^o  X )  .o  a )  +o  b
 )  =  C ) )   &    |-  Y  =  ( 1st `  P )   &    |-  Z  =  ( 2nd `  P )   =>    |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
 
Theoremcantnf 7411* The Cantor Normal Form theorem. The function  ( A CNF  B ), which maps a finitely supported function from  B to  A to the sum  ( ( A  ^o  f ( a 1 ) )  o.  a 1 )  +o  ( ( A  ^o  f ( a 2 ) )  o.  a 2 )  +o 
... over all indexes  a  <  B such that  f ( a ) is nonzero, is an order isomorphism from the ordering  T of finitely supported functions to the set  ( A  ^o  B
) under the natural order. Setting 
A  =  om and letting  B be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 7395, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
 
Theoremoemapwe 7412* The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternative definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
 
Theoremcantnffval2 7413* An alternative definition of df-cnf 7379 which relies on cantnf 7411. (Note that although the use of  S seems self-referential, one can use cantnfdm 7381 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S ) )
 
Theoremcantnff1o 7414 Simplify the isomorphism of cantnf 7411 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  ( A CNF  B ) : S -1-1-onto-> ( A  ^o  B ) )
 
Theoremmapfien 7415* A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } ) )  e. 
 Fin }   &    |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } )
 )  e.  Fin }   &    |-  W  =  ( G `  Z )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ph  ->  G : B -1-1-onto-> D )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  D  e.  _V )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 f  e.  S  |->  ( G  o.  ( f  o.  F ) ) ) : S -1-1-onto-> T )
 
Theoremwemapwe 7416* Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( z R w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  U  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } )
 )  e.  Fin }   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  S  We  B )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  F  = OrdIso ( R ,  A )   &    |-  G  = OrdIso ( S ,  B )   &    |-  Z  =  ( G `
  (/) )   =>    |-  ( ph  ->  T  We  U )
 
Theoremoef1o 7417* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption 
( F `  (/) )  =  (/) can be discharged using fveqf1o 5822.) (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  F : A -1-1-onto-> C )   &    |-  ( ph  ->  G : B -1-1-onto-> D )   &    |-  ( ph  ->  A  e.  ( On  \  1o ) )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  C  e.  On )   &    |-  ( ph  ->  D  e.  On )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  K  =  ( y  e.  { x  e.  ( A  ^m  B )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( F  o.  (
 y  o.  `' G ) ) )   &    |-  H  =  ( ( ( C CNF 
 D )  o.  K )  o.  `' ( A CNF 
 B ) )   =>    |-  ( ph  ->  H : ( A  ^o  B ) -1-1-onto-> ( C  ^o  D ) )
 
Theoremcnfcomlem 7418* Lemma for cnfcom 7419. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  ( ph  ->  I  e.  dom  G )   &    |-  ( ph  ->  O  e.  ( om  ^o  ( G `  I ) ) )   &    |-  ( ph  ->  ( T `  I ) : ( H `  I ) -1-1-onto-> O )   =>    |-  ( ph  ->  ( T `  suc  I ) : ( H `  suc  I ) -1-1-onto-> ( ( om  ^o  ( G `  I ) )  .o  ( F `
  ( G `  I ) ) ) )
 
Theoremcnfcom 7419* Any ordinal  B is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  ( ph  ->  I  e.  dom  G )   =>    |-  ( ph  ->  ( T `  suc  I ) : ( H `  suc  I
 )
 -1-1-onto-> ( ( om  ^o  ( G `  I ) )  .o  ( F `
  ( G `  I ) ) ) )
 
Theoremcnfcom2lem 7420* Lemma for cnfcom2 7421. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  (/)  e.  B )   =>    |-  ( ph  ->  dom  G  =  suc  U. dom  G )
 
Theoremcnfcom2 7421* Any nonzero ordinal  B is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  (/)  e.  B )   =>    |-  ( ph  ->  ( T `  dom  G ) : B -1-1-onto-> ( ( om  ^o  W )  .o  ( F `  W ) ) )
 
Theoremcnfcom3lem 7422* Lemma for cnfcom3 7423. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  om  C_  B )   =>    |-  ( ph  ->  W  e.  ( On  \  1o ) )
 
Theoremcnfcom3 7423* Any infinite ordinal  B is equinumerous to a power of  om. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 7425.) (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  om  C_  B )   &    |-  X  =  ( u  e.  ( F `
  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( F `  W )  .o  v )  +o  u ) )   &    |-  Y  =  ( u  e.  ( F `  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( om  ^o  W )  .o  u )  +o  v ) )   &    |-  N  =  ( ( X  o.  `' Y )  o.  ( T `  dom  G ) )   =>    |-  ( ph  ->  N : B -1-1-onto-> ( om  ^o  W ) )
 
Theoremcnfcom3clem 7424* Lemma for cnfcom3c 7425. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  F  =  ( `' ( om CNF  A ) `  b )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  X  =  ( u  e.  ( F `
  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( F `  W )  .o  v )  +o  u ) )   &    |-  Y  =  ( u  e.  ( F `  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( om  ^o  W )  .o  u )  +o  v ) )   &    |-  N  =  ( ( X  o.  `' Y )  o.  ( T `  dom  G ) )   &    |-  L  =  ( b  e.  ( om  ^o  A )  |->  N )   =>    |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( g `  b ) : b -1-1-onto-> ( om  ^o  w ) ) )
 
Theoremcnfcom3c 7425* Wrap the construction of cnfcom3 7423 into an existence quantifier. For any  om  C_  b, there is a bijection from  b to some power of  om. Furthermore, this bijection is canonical , which means that we can find a single function 
g which will give such bijections for every  b less than some arbitrarily large bound  A. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( g `  b ) : b -1-1-onto-> ( om  ^o  w ) ) )
 
2.6.4  Transitive closure
 
Theoremtrcl 7426* For any set  A, show the properties of its transitive closure  C. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 7427 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   &    |-  F  =  ( rec ( ( z  e.  _V  |->  ( z  u.  U. z
 ) ) ,  A )  |`  om )   &    |-  C  =  U_ y  e.  om  ( F `  y )   =>    |-  ( A  C_  C  /\  Tr  C  /\  A. x ( ( A  C_  x  /\  Tr  x ) 
 ->  C  C_  x )
 )
 
Theoremtz9.1 7427* Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7426 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself?

(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)

 |-  A  e.  _V   =>    |-  E. x ( A  C_  x  /\  Tr  x  /\  A. y
 ( ( A  C_  y  /\  Tr  y ) 
 ->  x  C_  y ) )
 
Theoremtz9.1c 7428* Alternative expression for the existence of transitive closures tz9.1 7427: the intersection of all transitive sets containing  A is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   =>    |-  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V
 
Theoremepfrs 7429* The strong form of the Axiom of Regularity (no sethood requirement on  A), with the axiom itself present as an antecedent. See also zfregs 7430. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  ( (  _E  Fr  A  /\  A  =/=  (/) )  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfregs 7430* The strong form of the Axiom of Regularity, which does not require that  A be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 7429. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfregs2 7431* Alternate strong form of the Axiom of Regularity. Not every element of a non-empty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
 |-  ( A  =/=  (/)  ->  -.  A. x  e.  A  E. y
 ( y  e.  A  /\  y  e.  x ) )
 
Theoremen3lplem1 7432* Lemma for en3lp 7434. (Contributed by Alan Sare, 28-Oct-2011.)
 |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) 
 ->  ( x  =  A  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
 
Theoremen3lplem2 7433* Lemma for en3lp 7434. (Contributed by Alan Sare, 28-Oct-2011.)
 |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) 
 ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
 )  =/=  (/) ) )
 
Theoremen3lp 7434 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 28937 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
 
Theoremsetind 7435* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
 |-  ( A. x ( x  C_  A  ->  x  e.  A )  ->  A  =  _V )
 
Theoremsetind2 7436 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
 |-  ( ~P A  C_  A  ->  A  =  _V )
 
Syntaxctc 7437 Extend class notation to include the transitive closure function.
 class  TC
 
Definitiondf-tc 7438* The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |- 
 TC  =  ( x  e.  _V  |->  |^| { y  |  ( x  C_  y  /\  Tr  y ) }
 )
 
Theoremtcvalg 7439* Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 7355; see tz9.1 7427.) (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  ( A  e.  V  ->  ( TC `  A )  =  |^| { x  |  ( A  C_  x  /\  Tr  x ) }
 )
 
Theoremtcid 7440 Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  ( A  e.  V  ->  A  C_  ( TC `  A ) )
 
Theoremtctr 7441 Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |- 
 Tr  ( TC `  A )
 
Theoremtcmin 7442 Defining property of the transitive closure function: it is a subset of any transitive class containing  A. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  ( A  e.  V  ->  ( ( A  C_  B  /\  Tr  B ) 
 ->  ( TC `  A )  C_  B ) )
 
Theoremtc2 7443* A variant of the definition of the transitive closure function, using instead the smallest transitive set containing  A as a member, gives almost the same set, except that  A itself must be added because it is not usually a member of  ( TC `  A
) (and it is never a member if  A is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( ( TC
 `  A )  u. 
 { A } )  =  |^| { x  |  ( A  e.  x  /\  Tr  x ) }
 
Theoremtcsni 7444 The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  A  e.  _V   =>    |-  ( TC `  { A } )  =  ( ( TC `  A )  u.  { A } )
 
Theoremtcss 7445 The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( B  C_  A  ->  ( TC `  B )  C_  ( TC
 `  A ) )
 
Theoremtcel 7446 The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  A  e.  _V   =>    |-  ( B  e.  A  ->  ( TC `  B )  C_  ( TC
 `  A ) )
 
Theoremtcidm 7447 The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  ( TC `  ( TC `  A ) )  =  ( TC `  A )
 
Theoremtc0 7448 The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  ( TC `  (/) )  =  (/)
 
Theoremtc00 7449 The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  ( A  e.  V  ->  ( ( TC `  A )  =  (/)  <->  A  =  (/) ) )
 
2.6.5  Rank
 
Syntaxcr1 7450 Extend class definition to include the cumulative hierarchy of sets function.
 class  R1
 
Syntaxcrnk 7451 Extend class definition to include rank function.
 class  rank
 
Definitiondf-r1 7452 Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation ( R1). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 7479). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 7456, r1suc 7458, and r1lim 7460. Theorem r1val1 7474 shows a recursive definition that works for all values, and theorems r1val2 7525 and r1val3 7526 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477),  _V with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)
 |- 
 R1  =  rec (
 ( x  e.  _V  |->  ~P x ) ,  (/) )
 
Definitiondf-rank 7453* Define the rank function. See rankval 7504, rankval2 7506, rankval3 7528, or rankval4 7555 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function 
R1: given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 7521 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 7524. (Contributed by NM, 11-Oct-2003.)
 |- 
 rank  =  ( x  e.  _V  |->  |^| { y  e. 
 On  |  x  e.  ( R1 `  suc  y ) } )
 
Theoremr1funlim 7454 The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 7455 avoids ax-rep 4147.) (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( Fun  R1  /\  Lim 
 dom  R1 )
 
Theoremr1fnon 7455 The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |- 
 R1  Fn  On
 
Theoremr10 7456 Value of the cumulative hierarchy of sets function at  (/). Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |-  ( R1 `  (/) )  =  (/)
 
Theoremr1sucg 7457 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  dom  R1 
 ->  ( R1 `  suc  A )  =  ~P ( R1 `  A ) )
 
Theoremr1suc 7458 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |-  ( A  e.  On  ->  ( R1 `  suc  A )  =  ~P ( R1 `  A ) )
 
Theoremr1limg 7459* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  dom 
 R1  /\  Lim  A ) 
 ->  ( R1 `  A )  =  U_ x  e.  A  ( R1 `  x ) )
 
Theoremr1lim 7460* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  B  /\  Lim  A )  ->  ( R1 `  A )  =  U_ x  e.  A  ( R1 `  x ) )
 
Theoremr1fin 7461 The first  om levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
 |-  ( A  e.  om  ->  ( R1 `  A )  e.  Fin )
 
Theoremr1sdom 7462 Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)
 |-  ( ( A  e.  On  /\  B  e.  A )  ->  ( R1 `  B )  ~<  ( R1
 `  A ) )
 
Theoremr111 7463 The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
 |- 
 R1 : On -1-1-> _V
 
Theoremr1tr 7464 The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |- 
 Tr  ( R1 `  A )
 
Theoremr1tr2 7465 The union of a cumulative hierarchy of sets at ordinal  A is a subset of the hierarchy at  A. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.)
 |- 
 U. ( R1 `  A )  C_  ( R1
 `  A )
 
Theoremr1ordg 7466 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.)
 |-  ( B  e.  dom  R1 
 ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
 
Theoremr1ord3g 7467 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)
 |-  ( ( A  e.  dom 
 R1  /\  B  e.  dom 
 R1 )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
 
Theoremr1ord 7468 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
 
Theoremr1ord2 7469 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 22-Sep-2003.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
 
Theoremr1ord3 7470 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1
 `  B ) ) )
 
Theoremr1sssuc 7471 The value of the cumulative hierarchy of sets function is a subset of its value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.)
 |-  ( A  e.  On  ->  ( R1 `  A )  C_  ( R1 `  suc  A ) )
 
Theoremr1pwss 7472 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  ( R1 `  B )  ->  ~P A  C_  ( R1
 `  B ) )
 
Theoremr1sscl 7473 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  ( R1 `  B ) 
 /\  C  C_  A )  ->  C  e.  ( R1 `  B ) )
 
Theoremr1val1 7474* The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  dom  R1 
 ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
 `  x ) )
 
Theoremtz9.12lem1 7475* Lemma for tz9.12 7478. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   &    |-  F  =  ( z  e.  _V  |->  |^|
 { v  e.  On  |  z  e.  ( R1 `  v ) }
 )   =>    |-  ( F " A )  C_  On
 
Theoremtz9.12lem2 7476* Lemma for tz9.12 7478. (Contributed by NM, 22-Sep-2003.)
 |-  A  e.  _V   &    |-  F  =  ( z  e.  _V  |->  |^|
 { v  e.  On  |  z  e.  ( R1 `  v ) }
 )   =>    |- 
 suc  U. ( F " A )  e.  On
 
Theoremtz9.12lem3 7477* Lemma for tz9.12 7478. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   &    |-  F  =  ( z  e.  _V  |->  |^|
 { v  e.  On  |  z  e.  ( R1 `  v ) }
 )   =>    |-  ( A. x  e.  A  E. y  e. 
 On  x  e.  ( R1 `  y )  ->  A  e.  ( R1 ` 
 suc  suc  U. ( F " A ) ) )
 
Theoremtz9.12 7478* A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 7475 through tz9.12lem3 7477. (Contributed by NM, 22-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  E. y  e. 
 On  x  e.  ( R1 `  y )  ->  E. y  e.  On  A  e.  ( R1 `  y ) )
 
Theoremtz9.13 7479* Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.)
 |-  A  e.  _V   =>    |-  E. x  e. 
 On  A  e.  ( R1 `  x )
 
Theoremtz9.13g 7480* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 7479 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  e.  ( R1 `  x ) )
 
Theoremrankwflemb 7481* Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x ) )
 
Theoremrankf 7482 The domain and range of the  rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)
 |- 
 rank : U. ( R1 " On ) --> On
 
Theoremrankon 7483 The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 12-Sep-2013.)
 |-  ( rank `  A )  e.  On
 
Theoremr1elwf 7484 Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
 
Theoremrankvalb 7485* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7504 does not use Regularity, and so requires the assumption that  A is in the range of  R1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  A )  =  |^| { x  e. 
 On  |  A  e.  ( R1 `  suc  x ) } )
 
Theoremrankr1ai 7486 One direction of rankr1a 7524. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
 
Theoremrankvaln 7487 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 7501, unless  A is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |-  ( -.  A  e.  U. ( R1 " On )  ->  ( rank `  A )  =  (/) )
 
Theoremrankidb 7488 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  A  e.  ( R1
 `  suc  ( rank `  A ) ) )
 
Theoremrankdmr1 7489 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( rank `  A )  e.  dom  R1
 
Theoremrankr1ag 7490 A version of rankr1a 7524 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B ) 
 <->  ( rank `  A )  e.  B ) )
 
Theoremrankr1bg 7491 A relationship between rank and  R1. See rankr1ag 7490 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B ) 
 <->  ( rank `  A )  C_  B ) )
 
Theoremr1rankidb 7492 Any set is a subset of the hierarchy of its rank. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  A  C_  ( R1 `  ( rank `  A )
 ) )
 
Theoremr1elssi 7493 The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7494 that doesn't need  A to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  A  C_  U. ( R1 " On ) )
 
Theoremr1elss 7494 The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( A  e.  U. ( R1 " On ) 
 <->  A  C_  U. ( R1 " On ) )
 
Theorempwwf 7495 A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  <->  ~P A  e.  U. ( R1 " On ) )
 
Theoremsswf 7496 A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  C_  A )  ->  B  e.  U. ( R1 " On )
 )
 
Theoremsnwf 7497 A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  { A }  e.  U. ( R1 " On ) )
 
Theoremunwf 7498 A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 ) 
 <->  ( A  u.  B )  e.  U. ( R1 " On ) )
 
Theoremprwf 7499 An unordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  { A ,  B }  e.  U. ( R1 " On )
 )
 
Theoremopwf 7500 An ordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  <. A ,  B >.  e.  U. ( R1 " On ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >