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Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupeq2 7201 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R )
 )
 
Theoremnfsup 7202 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x R   =>    |-  F/_ x sup ( A ,  B ,  R )
 
Theoremsupmo 7203* Any class  B has at most one supremum in  A (where  R is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  E* x  e.  A ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupexd 7204 A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoremsupeu 7205* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  E! x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupval2 7206* Alternative expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  ( iota_ x  e.  A ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) ) )
 
Theoremeqsup 7207* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  (
 ( C  e.  A  /\  A. y  e.  B  -.  C R y  /\  A. y  e.  A  ( y R C  ->  E. z  e.  B  y R z ) ) 
 ->  sup ( B ,  A ,  R )  =  C ) )
 
Theoremeqsupd 7208* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  C  e.  A )   &    |-  (
 ( ph  /\  y  e.  B )  ->  -.  C R y )   &    |-  (
 ( ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. z  e.  B  y R z )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
 
Theoremsupcl 7209* A supremum belongs to its base class (closure law). See also supub 7210 and suplub 7211. (Contributed by NM, 12-Oct-2004.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
 
Theoremsupub 7210* A supremum is an upper bound. See also supcl 7209 and suplub 7211.

This proof demonstrates how to expand an iota-based definition (df-iota 5219) using riotacl2 6318.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  ( C  e.  B  ->  -. 
 sup ( B ,  A ,  R ) R C ) )
 
Theoremsuplub 7211* A supremum is the least upper bound. See also supcl 7209 and supub 7210. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  (
 ( C  e.  A  /\  C R sup ( B ,  A ,  R ) )  ->  E. z  e.  B  C R z ) )
 
Theoremsuplub2 7212* Bidirectional form of suplub 7211. (Contributed by Mario Carneiro, 6-Sep-2014.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ( ph  /\  C  e.  A ) 
 ->  ( C R sup ( B ,  A ,  R )  <->  E. z  e.  B  C R z ) )
 
Theoremsupnub 7213* An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  (
 ( C  e.  A  /\  A. z  e.  B  -.  C R z ) 
 ->  -.  C R sup ( B ,  A ,  R ) ) )
 
Theoremsupex 7214 A supremum is a set. (Contributed by NM, 22-May-1999.)
 |-  R  Or  A   =>    |-  sup ( B ,  A ,  R )  e.  _V
 
Theoremsupmaxlem 7215* A set that contains a greatest element satisfies the antecedent in supremum theorems. This allows  sup ( A ,  B ,  R ) to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  ( ( C  e.  A  /\  C  e.  B  /\  A. z  e.  B  -.  C R z ) 
 ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupmax 7216* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B )  ->  -.  C R y )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
 
Theoremfisup2g 7217* A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  E. x  e.  B  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremfisupcl 7218 A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)
 |-  ( ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  sup ( B ,  A ,  R )  e.  B )
 
Theoremsuppr 7219 The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A ) 
 ->  sup ( { B ,  C } ,  A ,  R )  =  if ( C R B ,  B ,  C )
 )
 
Theoremsupsn 7220 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
 
Theoremsupisolem 7221* Lemma for supiso 7223. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   =>    |-  ( ( ph  /\  D  e.  A ) 
 ->  ( ( A. y  e.  C  -.  D R y  /\  A. y  e.  A  ( y R D  ->  E. z  e.  C  y R z ) )  <->  ( A. w  e.  ( F " C )  -.  ( F `  D ) S w 
 /\  A. w  e.  B  ( w S ( F `
  D )  ->  E. v  e.  ( F " C ) w S v ) ) ) )
 
Theoremsupisoex 7222* Lemma for supiso 7223. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   =>    |-  ( ph  ->  E. u  e.  B  ( A. w  e.  ( F " C )  -.  u S w 
 /\  A. w  e.  B  ( w S u  ->  E. v  e.  ( F " C ) w S v ) ) )
 
Theoremsupiso 7223* Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   &    |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  sup ( ( F " C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
 
2.4.38  Ordinal isomorphism, Hartog's theorem
 
Syntaxcoi 7224 Extend class definition to include the canonical order isomorphism to an ordinal.
 class OrdIso ( R ,  A )
 
Definitiondf-oi 7225* Define the canonical order isomorphism from the well-order  R on  A to an ordinal. (Contributed by Mario Carneiro, 23-May-2015.)
 |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e. 
 _V  |->  ( iota_ v  e. 
 { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e.  _V  |->  ( iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) ) " x ) z R t } ) ,  (/) )
 
Theoremdfoi 7226* Rewrite df-oi 7225 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  F  = recs ( G )   =>    |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  ( F  |`  { x  e. 
 On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } ) ,  (/) )
 
Theoremoieq1 7227 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  ( R  =  S  -> OrdIso ( R ,  A )  = OrdIso ( S ,  A ) )
 
Theoremoieq2 7228 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  ( A  =  B  -> OrdIso ( R ,  A )  = OrdIso ( R ,  B ) )
 
Theoremnfoi 7229 Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/_ xOrdIso ( R ,  A )
 
Theoremordiso2 7230 Generalize ordiso 7231 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  Ord  B ) 
 ->  A  =  B )
 
Theoremordiso 7231* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B 
 <-> 
 E. f  f  Isom  _E 
 ,  _E  ( A ,  B ) ) )
 
Theoremordtypecbv 7232* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   =>    |- recs
 ( ( f  e. 
 _V  |->  ( iota_ s  e. 
 { y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e.  { y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )  =  F
 
Theoremordtypelem1 7233* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O  =  ( F  |`  T ) )
 
Theoremordtypelem2 7234* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  Ord  T )
 
Theoremordtypelem3 7235* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ( ph  /\  M  e.  ( T  i^i  dom  F ) )  ->  ( F `  M )  e. 
 { v  e.  { w  e.  A  |  A. j  e.  ( F " M ) j R w }  |  A. u  e.  { w  e.  A  |  A. j  e.  ( F " M ) j R w }  -.  u R v } )
 
Theoremordtypelem4 7236* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O : ( T  i^i  dom 
 F ) --> A )
 
Theoremordtypelem5 7237* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  ( Ord  dom  O  /\  O : dom  O --> A ) )
 
Theoremordtypelem6 7238* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ( ph  /\  M  e.  dom  O )  ->  ( N  e.  M  ->  ( O `  N ) R ( O `  M ) ) )
 
Theoremordtypelem7 7239* Lemma for ordtype 7247. 
ran  O is an initial segment of  A under the well-order  R. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ( ( ph  /\  N  e.  A ) 
 /\  M  e.  dom  O )  ->  ( ( O `  M ) R N  \/  N  e.  ran 
 O ) )
 
Theoremordtypelem8 7240* Lemma for ordtype 7247. (Contributed by Mario Carneiro, 17-Oct-2009.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
 
Theoremordtypelem9 7241* Lemma for ordtype 7247. Either the function OrdIso is an isomorphism onto all of  A, or OrdIso is not a set, which by oif 7245 implies that either  ran  O 
C_  A is a proper class or  dom  O  =  On. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   &    |-  ( ph  ->  O  e.  _V )   =>    |-  ( ph  ->  O 
 Isom  _E  ,  R  ( dom  O ,  A ) )
 
Theoremordtypelem10 7242* Lemma for ordtype 7247. Using ax-rep 4131, exclude the possibility that  O is a proper class and does not enumerate all of 
A. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
 
Theoremoi0 7243 Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( -.  ( R  We  A  /\  R Se  A )  ->  F  =  (/) )
 
Theoremoicl 7244 The order type of the well-order  R on  A is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  Ord  dom  F
 
Theoremoif 7245 The order isomorphism of the well-order  R on  A is a function. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  F : dom  F --> A
 
Theoremoiiso2 7246 The order isomorphism of the well-order  R on  A is an isomorphism onto  ran  O (which is a subset of  A by oif 7245). (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E 
 ,  R  ( dom 
 F ,  ran  F ) )
 
Theoremordtype 7247 For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E 
 ,  R  ( dom 
 F ,  A ) )
 
Theoremoiiniseg 7248  ran  F is an initial segment of  A under the well-order  R. (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( N  e.  A  /\  M  e.  dom  F ) )  ->  ( ( F `  M ) R N  \/  N  e.  ran 
 F ) )
 
Theoremordtype2 7249 For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto  A isomorphically. Otherwise  F is a proper class, which implies that either 
ran  F  C_  A is a proper class or  dom  F  =  On. This weak version of ordtype 7247 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( R  We  A  /\  R Se  A 
 /\  F  e.  _V )  ->  F  Isom  _E  ,  R  ( dom  F ,  A ) )
 
Theoremoiexg 7250 The order isomorphism on an set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( A  e.  V  ->  F  e.  _V )
 
Theoremoion 7251 The order type of the well-order  R on  A is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( A  e.  V  ->  dom  F  e.  On )
 
Theoremoiiso 7252 The order isomorphism of the well-order  R on  A is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( A  e.  V  /\  R  We  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A ) )
 
Theoremoien 7253 The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( A  e.  V  /\  R  We  A )  ->  dom  F  ~~  A )
 
Theoremoieu 7254 Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( R  We  A  /\  R Se  A )  ->  ( ( Ord  B  /\  G  Isom  _E 
 ,  R  ( B ,  A ) )  <-> 
 ( B  =  dom  F 
 /\  G  =  F ) ) )
 
Theoremoismo 7255 When  A is a subclass of  On,  F is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of  A). The proof avoids ax-rep 4131 (the second statement is trivial under ax-rep 4131). (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  F  = OrdIso (  _E 
 ,  A )   =>    |-  ( A  C_  On  ->  ( Smo  F  /\  ran  F  =  A ) )
 
Theoremoiid 7256 The order type of an ordinal under the  e. order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  ( Ord  A  -> OrdIso (  _E  ,  A )  =  (  _I  |`  A ) )
 
Theoremhartogslem1 7257* Lemma for hartogs 7259. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  F  =  { <. r ,  y >.  |  ( ( ( dom  r  C_  A  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
 ) )  /\  (
 r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }   &    |-  R  =  { <. s ,  t >.  | 
 E. w  e.  y  E. z  e.  y  ( ( s  =  ( f `  w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }   =>    |-  ( dom  F  C_ 
 ~P ( A  X.  A )  /\  Fun  F  /\  ( A  e.  V  ->  ran  F  =  { x  e.  On  |  x  ~<_  A } ) )
 
Theoremhartogslem2 7258* Lemma for hartogs 7259. (Contributed by Mario Carneiro, 14-Jan-2013.)
 |-  F  =  { <. r ,  y >.  |  ( ( ( dom  r  C_  A  /\  (  _I  |`  dom  r )  C_  r  /\  r  C_  ( dom  r  X.  dom  r
 ) )  /\  (
 r  \  _I  )  We  dom  r )  /\  y  =  dom OrdIso ( ( r  \  _I  ) ,  dom  r ) ) }   &    |-  R  =  { <. s ,  t >.  | 
 E. w  e.  y  E. z  e.  y  ( ( s  =  ( f `  w )  /\  t  =  ( f `  z ) )  /\  w  _E  z ) }   =>    |-  ( A  e.  V  ->  { x  e. 
 On  |  x  ~<_  A }  e.  _V )
 
Theoremhartogs 7259* Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 8185- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
 
Theoremwofib 7260 The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( R  Or  A  /\  A  e.  Fin )  <->  ( R  We  A  /\  `' R  We  A ) )
 
Theoremwemaplem1 7261* Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ( P  e.  V  /\  Q  e.  W )  ->  ( P T Q 
 <-> 
 E. a  e.  A  ( ( P `  a ) S ( Q `  a ) 
 /\  A. b  e.  A  ( b R a 
 ->  ( P `  b
 )  =  ( Q `
  b ) ) ) ) )
 
Theoremwemaplem2 7262* Lemma for wemapso 7266. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  P  e.  ( B  ^m  A ) )   &    |-  ( ph  ->  X  e.  ( B  ^m  A ) )   &    |-  ( ph  ->  Q  e.  ( B  ^m  A ) )   &    |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  S  Po  B )   &    |-  ( ph  ->  a  e.  A )   &    |-  ( ph  ->  ( P `  a ) S ( X `  a ) )   &    |-  ( ph  ->  A. c  e.  A  ( c R a  ->  ( P `  c )  =  ( X `  c ) ) )   &    |-  ( ph  ->  b  e.  A )   &    |-  ( ph  ->  ( X `  b ) S ( Q `  b ) )   &    |-  ( ph  ->  A. c  e.  A  ( c R b 
 ->  ( X `  c
 )  =  ( Q `
  c ) ) )   =>    |-  ( ph  ->  P T Q )
 
Theoremwemaplem3 7263* Lemma for wemapso 7266. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  P  e.  ( B  ^m  A ) )   &    |-  ( ph  ->  X  e.  ( B  ^m  A ) )   &    |-  ( ph  ->  Q  e.  ( B  ^m  A ) )   &    |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  S  Po  B )   &    |-  ( ph  ->  P T X )   &    |-  ( ph  ->  X T Q )   =>    |-  ( ph  ->  P T Q )
 
Theoremwemappo 7264* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Po  B ) 
 ->  T  Po  ( B 
 ^m  A ) )
 
Theoremwemapso2lem 7265* Lemma for wemapso 7266. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  U  C_  ( B  ^m  A )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  S  Or  B )   &    |-  (
 ( ph  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b
 ) )  ->  E. c  e.  dom  ( a  \  b ) A. d  e.  dom  ( a  \  b )  -.  d R c )   =>    |-  ( ph  ->  T  Or  U )
 
Theoremwemapso 7266* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ( A  e.  V  /\  R  We  A  /\  S  Or  B ) 
 ->  T  Or  ( B 
 ^m  A ) )
 
Theoremwemapso2 7267* An alternative to having a well-order on  R in wemapso 7266 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  U  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } )
 )  e.  Fin }   =>    |-  (
 ( A  e.  V  /\  R  Or  A  /\  S  Or  B )  ->  T  Or  U )
 
Theoremcard2on 7268* Proof that the alternate definition cardval2 7624 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
 |- 
 { x  e.  On  |  x  ~<  A }  e.  On
 
Theoremcard2inf 7269* The definition cardval2 7624 has the curious property that for non-numerable sets (for which ndmfv 5552 yields  (/)), it still evaluates to a non-empty set, and indeed it contains  om. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( -.  E. y  e.  On  y  ~~  A  ->  om  C_  { x  e.  On  |  x  ~<  A } )
 
2.4.39  Hartogs function, order types, weak dominance
 
Syntaxchar 7270 Class symbol for the Hartogs/cardinal successor function.
 class har
 
Syntaxcwdom 7271 Class symbol for the weak dominance relation.
 class  ~<_*
 
Definitiondf-har 7272* Define the Hartogs function , which maps all sets to the smallest ordinal that cannot be injected into the given set. In the important special case where  x is an ordinal, this is the cardinal successor operation.

Traditionally, the Hartogs number of a set is written  aleph ( X ) and the cardinal successor 
X  +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 7573.

Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.)

 |- har 
 =  ( x  e. 
 _V  |->  { y  e.  On  |  y  ~<_  x }
 )
 
Definitiondf-wdom 7273* A set is weakly dominated by a "larger" set iff the "larger" set can be mapped onto the "smaller" set or the smaller set is empty; equivalently if the smaller set can be placed into bijection with some partition of the larger set. When choice is assumed (as fodom 8149), this concides with the 1-1 defition df-dom 6865; however, it is not known whether this is a choice-equivalent or a strictly weaker form. Some discussion of this question can be found at http://boolesrings.org/asafk/2014/on-the-partition-principle/. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ~<_*  =  { <. x ,  y >.  |  ( x  =  (/)  \/  E. z  z : y -onto-> x ) }
 
Theoremharf 7274 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |- har : _V --> On
 
Theoremharcl 7275 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  (har `  X )  e.  On
 
Theoremharval 7276* Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  (har `  X )  =  { y  e.  On  |  y  ~<_  X }
 )
 
Theoremelharval 7277 The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( Y  e.  (har `  X )  <->  ( Y  e.  On  /\  Y  ~<_  X ) )
 
Theoremharndom 7278 The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
 |- 
 -.  (har `  X ) 
 ~<_  X
 
Theoremharword 7279 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  ( X  ~<_  Y  ->  (har `  X )  C_  (har `  Y ) )
 
Theoremrelwdom 7280 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |- 
 Rel  ~<_*
 
Theorembrwdom 7281* Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( Y  e.  V  ->  ( X  ~<_*  Y  <->  ( X  =  (/) 
 \/  E. z  z : Y -onto-> X ) ) )
 
Theorembrwdomi 7282* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  ->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) )
 
Theorembrwdomn0 7283* Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( X  =/=  (/)  ->  ( X 
 ~<_*  Y 
 <-> 
 E. z  z : Y -onto-> X ) )
 
Theorem0wdom 7284 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  -> 
 (/)  ~<_*  X )
 
Theoremfowdom 7285 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( ( F  e.  V  /\  F : Y -onto-> X )  ->  X  ~<_*  Y )
 
Theoremwdomref 7286 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  X  ~<_*  X )
 
Theorembrwdom2 7287* Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( Y  e.  V  ->  ( X  ~<_*  Y  <->  E. y  e.  ~P  Y E. z  z : y -onto-> X ) )
 
Theoremdomwdom 7288 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
 
Theoremwdomtr 7289 Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  ->  X  ~<_*  Z )
 
Theoremwdomen1 7290 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
 
Theoremwdomen2 7291 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )
 
Theoremwdompwdom 7292 Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y )
 
Theoremcanthwdom 7293 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7014, equivalent to canth 6294). (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 -.  ~P A  ~<_*  A
 
Theoremwdom2d 7294* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4131). (Contributed by Stefan O'Rear, 13-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  (
 ( ph  /\  x  e.  A )  ->  E. y  e.  B  x  =  X )   =>    |-  ( ph  ->  A  ~<_*  B )
 
Theoremwdomd 7295* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
 |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  E. y  e.  B  x  =  X )   =>    |-  ( ph  ->  A  ~<_*  B )
 
Theorembrwdom3 7296* Condition for weak dominance with a condition reminiscent of wdomd 7295. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( X  ~<_*  Y  <->  E. f A. x  e.  X  E. y  e.  Y  x  =  ( f `  y ) ) )
 
Theorembrwdom3i 7297* Weak dominance implies existance of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
 |-  ( X  ~<_*  Y  ->  E. f A. x  e.  X  E. y  e.  Y  x  =  ( f `  y ) )
 
Theoremunwdomg 7298 Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( A  ~<_*  B  /\  C  ~<_*  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~<_*  ( B  u.  D ) )
 
Theoremxpwdomg 7299 Weak dominance of a cross product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( A  ~<_*  B  /\  C  ~<_*  D )  ->  ( A  X.  C )  ~<_*  ( B  X.  D ) )
 
Theoremwdomima2g 7300 A set is weakly dominant over its image under any function. This version of wdomimag 7301 is stated so as to avoid ax-rep 4131. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A )
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