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Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremordtypelem1 7201* Lemma for ordtype 7215. (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 7202* Lemma for ordtype 7215. (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 7203* Lemma for ordtype 7215. (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 7204* Lemma for ordtype 7215. (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 7205* Lemma for ordtype 7215. (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 7206* Lemma for ordtype 7215. (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 7207* Lemma for ordtype 7215. 
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 7208* Lemma for ordtype 7215. (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 7209* Lemma for ordtype 7215. Either the function OrdIso is an isomorphism onto all of  A, or OrdIso is not a set, which by oif 7213 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 7210* Lemma for ordtype 7215. Using ax-rep 4105, 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 7211 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 7212 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 7213 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 7214 The order isomorphism of the well-order  R on  A is an isomorphism onto  ran  O (which is a subset of  A by oif 7213). (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 7215 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 7216  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 7217 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 7215 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 7218 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 7219 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 7220 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 7221 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 7222 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 7223 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 4105 (the second statement is trivial under ax-rep 4105). (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  F  = OrdIso (  _E 
 ,  A )   =>    |-  ( A  C_  On  ->  ( Smo  F  /\  ran  F  =  A ) )
 
Theoremoiid 7224 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 7225* Lemma for hartogs 7227. (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 7226* Lemma for hartogs 7227. (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 7227* 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 8153- 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 7228 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 7229* 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 7230* Lemma for wemapso 7234. 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 7231* Lemma for wemapso 7234. 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 7232* 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 7233* Lemma for wemapso 7234. (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 7234* 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 7235* An alternative to having a well-order on  R in wemapso 7234 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 7236* Proof that the alternate definition cardval2 7592 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
 |- 
 { x  e.  On  |  x  ~<  A }  e.  On
 
Theoremcard2inf 7237* The definition cardval2 7592 has the curious property that for non-numerable sets (for which ndmfv 5486 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 7238 Class symbol for the Hartogs/cardinal successor function.
 class har
 
Syntaxcwdom 7239 Class symbol for the weak dominance relation.
 class  ~<_*
 
Definitiondf-har 7240* 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 7541.

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 7241* 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 8117), this concides with the 1-1 defition df-dom 6833; 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 7242 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |- har : _V --> On
 
Theoremharcl 7243 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  (har `  X )  e.  On
 
Theoremharval 7244* 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 7245 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 7246 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 7247 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  ( X  ~<_  Y  ->  (har `  X )  C_  (har `  Y ) )
 
Theoremrelwdom 7248 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |- 
 Rel  ~<_*
 
Theorembrwdom 7249* 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 7250* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  ->  ( X  =  (/)  \/  E. z  z : Y -onto-> X ) )
 
Theorembrwdomn0 7251* 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 7252 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  -> 
 (/)  ~<_*  X )
 
Theoremfowdom 7253 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( ( F  e.  V  /\  F : Y -onto-> X )  ->  X  ~<_*  Y )
 
Theoremwdomref 7254 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  X  ~<_*  X )
 
Theorembrwdom2 7255* 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 7256 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
 
Theoremwdomtr 7257 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 7258 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
 
Theoremwdomen2 7259 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )
 
Theoremwdompwdom 7260 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 7261 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 6982, equivalent to canth 6260). (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 -.  ~P A  ~<_*  A
 
Theoremwdom2d 7262* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4105). (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 7263* 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 7264* Condition for weak dominance with a condition reminiscent of wdomd 7263. (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 7265* 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 7266 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 7267 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 7268 A set is weakly dominant over its image under any function. This version of wdomimag 7269 is stated so as to avoid ax-rep 4105. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A )
 
Theoremwdomimag 7269 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 7270 Lemma for unxpwdom 7271. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  X.  A )  ~~  ( B  u.  C )  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
 
Theoremunxpwdom 7271 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 7272 The Hartogs function is weakly dominated by  ~P ( X  X.  X
). This follows from a more precise analysis of the bound used in hartogs 7227 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 7273* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 6814 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 7274* 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 7277) 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 7279). A stronger version that works for proper classes is proved as zfregs 7382. (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 7275* 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 7276* 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 7277* 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 7382). (Contributed by NM, 26-Nov-1995.)
 |-  A  e.  _V   =>    |-  ( A  =/=  (/) 
 ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfreg2 7278* The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7277) 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 7279 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 7284 and efrirr 4346, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
 |- 
 -.  x  e.  x
 
Theoremelirr 7280 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 7281 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 7282 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 7283 Alternate proof of Russell's Paradox ru 2965, simplified using (indirectly) the Axiom of Regularity ax-reg 7274. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.)
 |- 
 { x  |  x  e/  x }  e/  _V
 
Theoremzfregfr 7284 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
 |- 
 _E  Fr  A
 
Theoremen2lp 7285 No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |- 
 -.  ( A  e.  B  /\  B  e.  A )
 
Theorempreleq 7286 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( ( A  e.  B  /\  C  e.  D )  /\  { A ,  B }  =  { C ,  D } )  ->  ( A  =  C  /\  B  =  D ) )
 
Theoremopthreg 7287 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7274 (via the preleq 7286 step). See df-op 3623 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremsuc11reg 7288 The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
 |-  ( suc  A  =  suc  B  <->  A  =  B )
 
Theoremdford2 7289* Assuming ax-reg 7274, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) )
 
2.5.2  Axiom of Infinity equivalents
 
Theoreminf0 7290* Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class " ran  ( rec (
( v  e.  _V  |->  suc  v ) ,  x
)  |`  om ) " exists, is a subset of its union, and contains a given set  x (and thus is non-empty). Thus it provides an example demonstrating that a set  y exists with the necessary properties demanded by ax-inf 7307. (Contributed by NM, 15-Oct-1996.)
 |- 
 om  e.  _V   =>    |- 
 E. y ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y
 ) ) )
 
Theoreminf1 7291 Variation of Axiom of Infinity (using zfinf 7308 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
 |- 
 E. x ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )   =>    |-  E. x ( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )
 
Theoreminf2 7292* Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7308 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
 |- 
 E. x ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )   =>    |-  E. x ( x  =/=  (/)  /\  x  C_ 
 U. x )
 
Theoreminf3lema 7293* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  ( G `  B )  <->  ( A  e.  x  /\  ( A  i^i  x )  C_  B )
 )
 
Theoreminf3lemb 7294* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F `  (/) )  =  (/)
 
Theoreminf3lemc 7295* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  om  ->  ( F `  suc  A )  =  ( G `  ( F `  A ) ) )
 
Theoreminf3lemd 7296* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  om  ->  ( F `  A )  C_  x )
 
Theoreminf3lem1 7297* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
 
Theoreminf3lem2 7298* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( A  e.  om  ->  ( F `  A )  =/=  x ) )
 
Theoreminf3lem3 7299* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7277. (Contributed by NM, 29-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( A  e.  om  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
 
Theoreminf3lem4 7300* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7304 for detailed description. (Contributed by NM, 29-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( A  e.  om  ->  ( F `  A ) 
 C.  ( F `  suc  A ) ) )
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