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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfisup2g 7101* 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 7102 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 7103 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 7104 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
 
Theoremsupisolem 7105* Lemma for supiso 7107. (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 7106* Lemma for supiso 7107. (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 7107* 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 ) ) )
 
2.4.36  Ordinal isomorphism, Hartog's theorem
 
Syntaxcoi 7108 Extend class definition to include the canonical order isomorphism to an ordinal.
 class OrdIso ( R ,  A )
 
Definitiondf-oi 7109* 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 7110* Rewrite df-oi 7109 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 7111 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  ( R  =  S  -> OrdIso ( R ,  A )  = OrdIso ( S ,  A ) )
 
Theoremoieq2 7112 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  ( A  =  B  -> OrdIso ( R ,  A )  = OrdIso ( R ,  B ) )
 
Theoremnfoi 7113 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 7114 Generalize ordiso 7115 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  Ord  B ) 
 ->  A  =  B )
 
Theoremordiso 7115* 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 7116* Lemma for ordtype 7131. (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 7117* Lemma for ordtype 7131. (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 7118* Lemma for ordtype 7131. (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 7119* Lemma for ordtype 7131. (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 7120* Lemma for ordtype 7131. (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 7121* Lemma for ordtype 7131. (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 7122* Lemma for ordtype 7131. (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 7123* Lemma for ordtype 7131. 
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 7124* Lemma for ordtype 7131. (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 7125* Lemma for ordtype 7131. Either the function OrdIso is an isomorphism onto all of  A, or OrdIso is not a set, which by oif 7129 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 7126* Lemma for ordtype 7131. Using ax-rep 4028, 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 7127 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 7128 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 7129 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 7130 The order isomorphism of the well-order  R on  A is an isomorphism onto  ran  O (which is a subset of  A by oif 7129). (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 7131 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 7132  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 7133 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 7131 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 7134 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 7135 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 7136 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 7137 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 7138 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 7139 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 4028 (the second statement is trivial under ax-rep 4028). (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  F  = OrdIso (  _E 
 ,  A )   =>    |-  ( A  C_  On  ->  ( Smo  F  /\  ran  F  =  A ) )
 
Theoremoiid 7140 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 7141* Lemma for hartogs 7143. (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 7142* Lemma for hartogs 7143. (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 7143* 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 8067- 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 7144 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 7145* 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 7146* Lemma for wemapso 7150. 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 7147* Lemma for wemapso 7150. 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 7148* 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 7149* Lemma for wemapso 7150. (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 7150* 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 7151* An alternative to having a well-order on  R in wemapso 7150 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 7152* Proof that the alternate definition cardval2 7508 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
 |- 
 { x  e.  On  |  x  ~<  A }  e.  On
 
Theoremcard2inf 7153* The definition cardval2 7508 has the curious property that for non-numerable sets (for which ndmfv 5405 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 } )
 
2.4.37  Hartogs function, order types, weak dominance
 
Syntaxchar 7154 Class symbol for the Hartogs/cardinal successor function.
 class har
 
Syntaxcwdom 7155 Class symbol for the weak dominance relation.
 class  ~<_*
 
Definitiondf-har 7156* 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 7457.

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 7157* 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 8033), this concides with the 1-1 defition df-dom 6751; 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 7158 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |- har : _V --> On
 
Theoremharcl 7159 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  (har `  X )  e.  On
 
Theoremharval 7160* 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 7161 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 7162 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 7163 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  ( X  ~<_  Y  ->  (har `  X )  C_  (har `  Y ) )
 
Theoremrelwdom 7164 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |- 
 Rel  ~<_*
 
Theorembrwdom 7165* 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 7166* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  ->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) )
 
Theorembrwdomn0 7167* 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 7168 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  -> 
 (/)  ~<_*  X )
 
Theoremfowdom 7169 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( ( F  e.  V  /\  F : Y -onto-> X )  ->  X  ~<_*  Y )
 
Theoremwdomref 7170 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  X  ~<_*  X )
 
Theorembrwdom2 7171* 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 7172 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
 
Theoremwdomtr 7173 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 7174 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
 
Theoremwdomen2 7175 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )
 
Theoremwdompwdom 7176 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 7177 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 6899, equivalent to canth 6178). (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 -.  ~P A  ~<_*  A
 
Theoremwdom2d 7178* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4028). (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 7179* 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 7180* Condition for weak dominance with a condition reminiscent of wdomd 7179. (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 7181* 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 7182 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 7183 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 7184 A set is weakly dominant over its image under any function. This version of wdomimag 7185 is stated so as to avoid ax-rep 4028. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A )
 
Theoremwdomimag 7185 A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( Fun  F  /\  A  e.  V ) 
 ->  ( F " A ) 
 ~<_*  A )
 
Theoremunxpwdom2 7186 Lemma for unxpwdom 7187. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  X.  A )  ~~  ( B  u.  C )  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
 
Theoremunxpwdom 7187 If a cross product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  X.  A )  ~<_  ( B  u.  C )  ->  ( A 
 ~<_*  B  \/  A  ~<_  C ) )
 
Theoremharwdom 7188 The Hartogs function is weakly dominated by  ~P ( X  X.  X
). This follows from a more precise analysis of the bound used in hartogs 7143 to prove that  (har `  X ) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )
 
Theoremixpiunwdom 7189* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 6732 this shows that  U_ x  e.  A B and  X_ x  e.  A B have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( A  e.  V  /\  U_ x  e.  A  B  e.  W  /\  X_ x  e.  A  B  =/= 
 (/) )  ->  U_ x  e.  A  B  ~<_*  ( X_ x  e.  A  B  X.  A ) )
 
2.5  ZF Set Theory - add the Axiom of Regularity
 
2.5.1  Introduce the Axiom of Regularity
 
Axiomax-reg 7190* Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 7193) that every non-empty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 7195). A stronger version that works for proper classes is proved as zfregs 7298. (Contributed by NM, 14-Aug-1993.)
 |-  ( E. y  y  e.  x  ->  E. y
 ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
 
Theoremaxreg2 7191* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
 |-  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremzfregcl 7192* The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A )
 
Theoremzfreg 7193* The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." There is also a "strong form," not requiring that  A be a set, that can be proved with more difficulty (see zfregs 7298). (Contributed by NM, 26-Nov-1995.)
 |-  A  e.  _V   =>    |-  ( A  =/=  (/) 
 ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfreg2 7194* The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7193) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A  =/=  (/) 
 ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
 
Theoremelirrv 7195 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 7200 and efrirr 4267, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
 |- 
 -.  x  e.  x
 
Theoremelirr 7196 No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |- 
 -.  A  e.  A
 
Theoremsucprcreg 7197 A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.)
 |-  ( -.  A  e.  _V  <->  suc 
 A  =  A )
 
Theoremruv 7198 The Russell class is equal to the universe  _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
 |- 
 { x  |  x  e/  x }  =  _V
 
TheoremruALT 7199 Alternate proof of Russell's Paradox ru 2920, simplified using (indirectly) the Axiom of Regularity ax-reg 7190. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.)
 |- 
 { x  |  x  e/  x }  e/  _V
 
Theoremzfregfr 7200 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
 |- 
 _E  Fr  A
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