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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrins 23601* Founded Induction Schema. If a property passes from all elements less than  y of a founded class  A to  y itself (induction hypothesis), then the property holds for all elements of  A. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremfrins2fg 23602* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   &    |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   =>    |-  ( ( R  Fr  A  /\  R Se  A )  ->  A. y  e.  A  ph )
 
Theoremfrins2f 23603* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   &    |-  ( y  e.  A  ->  ( A. z  e.  Pred  ( R ,  A ,  y
 ) ps  ->  ph )
 )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremfrins2g 23604* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   =>    |-  ( ( R  Fr  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremfrins2 23605* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremfrins3 23606* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( B  e.  A  ->  ch )
 
16.7.18  Ordering Ordinal Sequences
 
Theoremorderseqlem 23607* Lemma for poseq 23608 and soseq 23609. The function value of a sequene is either in  A or null. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  F  =  { f  |  E. x  e.  On  f : x --> A }   =>    |-  ( G  e.  F  ->  ( G `  X )  e.  ( A  u.  { (/) } )
 )
 
Theoremposeq 23608* A partial ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  R  Po  ( A  u.  { (/)
 } )   &    |-  F  =  {
 f  |  E. x  e.  On  f : x --> A }   &    |-  S  =  { <. f ,  g >.  |  ( ( f  e.  F  /\  g  e.  F )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
 )  =  ( g `
  y )  /\  ( f `  x ) R ( g `  x ) ) ) }   =>    |-  S  Po  F
 
Theoremsoseq 23609* A linear ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  R  Or  ( A  u.  { (/)
 } )   &    |-  F  =  {
 f  |  E. x  e.  On  f : x --> A }   &    |-  S  =  { <. f ,  g >.  |  ( ( f  e.  F  /\  g  e.  F )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
 )  =  ( g `
  y )  /\  ( f `  x ) R ( g `  x ) ) ) }   &    |-  -.  (/)  e.  A   =>    |-  S  Or  F
 
16.7.19  Well-founded recursion
 
Theoremwfr3g 23610* Functions defined by well founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 ( ( R  We  A  /\  R Se  A ) 
 /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
 ) ) ) ) 
 ->  F  =  G )
 
Theoremwfrlem1 23611* Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions  B. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  B  =  { g  |  E. z ( g  Fn  z  /\  (
 z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z ) 
 /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w ) ) ) ) }
 
Theoremwfrlem2 23612* Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( g  e.  B  ->  Fun  g )
 
Theoremwfrlem3 23613* Lemma for well-founded recursion. An acceptable function's domain is a subset of  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( g  e.  B  ->  dom  g  C_  A )
 
Theoremwfrlem4 23614* Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another one. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  B  =  {
 f  |  E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x )  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i  dom  h )  /\  A. a  e.  ( dom  g  i^i 
 dom  h ) ( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a
 )  =  ( G `
  ( ( g  |`  ( dom  g  i^i 
 dom  h ) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
 
Theoremwfrlem5 23615* Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremwfrlem6 23616* Lemma for well-founded recursion. The union of all acceptable functions is a relationship. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Rel  F
 
Theoremwfrlem7 23617* Lemma for well-founded recursion. The domain of  F is a subclass of  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 dom  F  C_  A
 
Theoremwfrlem8 23618* Lemma for well-founded recursion. Compute the prececessor class for an  R minimal element of  ( A  \  dom  F ). (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( Pred ( R ,  ( A  \  dom  F ) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  =  Pred ( R ,  dom  F ,  X ) )
 
Theoremwfrlem9 23619* Lemma for well-founded recursion. If  X  e.  dom  F, then its predecessor class is a subset of  dom  F. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( X  e.  dom  F 
 ->  Pred ( R ,  A ,  X )  C_ 
 dom  F )
 
Theoremwfrlem10 23620* Lemma for well-founded recursion. When  z is an  R minimal element of  ( A  \  dom  F ), then its predecessor class is equal to  dom  F. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  B  =  {
 f  |  E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x )  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( ( z  e.  ( A  \  dom  F )  /\  Pred ( R ,  ( A  \ 
 dom  F ) ,  z
 )  =  (/) )  ->  Pred ( R ,  A ,  z )  =  dom  F )
 
Theoremwfrlem11 23621* Lemma for well-founded recursion. The union of all acceptable functions is a function. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Fun  F
 
Theoremwfrlem12 23622* Lemma for well-founded recursion. Here, we compute the value of  F (the union of all acceptable functions). (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( y  e.  dom  F 
 ->  ( F `  y
 )  =  ( G `
  ( F  |`  Pred ( R ,  A ,  y ) ) ) )
 
Theoremwfrlem13 23623* Lemma for well-founded recursion. From here through wfrlem16 23626, we aim to prove that  dom  F  =  A. We do this by supposing that there is an element  z of  A that is not in  dom  F. We then define  C by extending  dom  F with the appropriate value at  z. We then show that  z cannot be an  R minimal element of  ( A  \  dom  F ), meaning that  ( A  \  dom  F ) must be empty, so  dom  F  =  A. Here, we show that  C is a function extending the domain of  F by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  (
 z  e.  ( A 
 \  dom  F )  ->  C  Fn  ( dom 
 F  u.  { z } ) )
 
Theoremwfrlem14 23624* Lemma for well-founded recursion. Compute the value of  C. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  (
 z  e.  ( A 
 \  dom  F )  ->  ( y  e.  ( dom  F  u.  { z } )  ->  ( C `
  y )  =  ( G `  ( C  |`  Pred ( R ,  A ,  y )
 ) ) ) )
 
Theoremwfrlem15 23625* Lemma for well-founded recursion. When  z is  R minimal,  C is an acceptable function. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  (
 ( z  e.  ( A  \  dom  F ) 
 /\  Pred ( R ,  ( A  \  dom  F ) ,  z )  =  (/) )  ->  C  e.  B )
 
Theoremwfrlem16 23626* Lemma for well-founded recursion. If 
z is  R minimal in  ( A  \  dom  F ), then  C is acceptable and thus a subset of  F, but  dom  C is bigger than  dom  F. Thus, 
z cannot be minimal, so  ( A  \  dom  F ) must be empty, and (due to wfrlem7 23617),  dom  F  =  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  dom  F  =  A
 
Theoremwfr1 23627* The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function  G and a class of "acceptable" functions  B. Then, using a base class  A and a well-ordering  R of  A, we define a function  F. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of  F. We begin by showing that  F is a function over  A. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  F  Fn  A
 
Theoremwfr2 23628* The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of  F at any  z  e.  A is  G recursively applied to all "previous" values of  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( z  e.  A  ->  ( F `  z
 )  =  ( G `
  ( F  |`  Pred ( R ,  A ,  z ) ) ) )
 
Theoremwfr2c 23629* Generalize wfr2 23628 to class arguments. (Contributed by Scott Fenton, 6-Aug-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( X  e.  A  ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
 
Theoremwfr3 23630* The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that  F is unique. We do this by showing that any function  H with the same properties we proved of  F in wfr1 23627 and wfr2 23628 is identical to  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A ,  z ) ) ) )  ->  F  =  H )
 
16.7.20  Transfinite recursion via Well-founded recursion
 
TheoremtfrALTlem 23631* Lemma for deriving transfinite recursion from well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.)
 |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  y ) ) ) }  =  { f  |  E. x ( f  Fn  x  /\  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 (  _E  ,  On ,  y ) ) ) ) }
 
Theoremtfr1ALT 23632* tfr1 6367 via well-founded recursion. (Contributed by Scott Fenton, 17-Aug-1994.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  A  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  F  =  U. A   =>    |-  F  Fn  On
 
Theoremtfr2ALT 23633* tfr2 6368 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  A  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  F  =  U. A   =>    |-  ( z  e.  On  ->  ( F `  z
 )  =  ( G `
  ( F  |`  z ) ) )
 
Theoremtfr3ALT 23634* tfr3 6369 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  A  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  F  =  U. A   =>    |-  ( ( B  Fn  On  /\  A. x  e. 
 On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
 
16.7.21  Founded Recursion
 
Theoremfrr3g 23635* Functions defined by founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  Fr  A  /\  R Se  A ) 
 /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  (
 y H ( F  |`  Pred ( R ,  A ,  y )
 ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( y H ( G  |`  Pred ( R ,  A ,  y ) ) ) ) )  ->  F  =  G )
 
Theoremfrrlem1 23636* Lemma for founded recursion. The final item we are interested in is the union of acceptable functions  B. This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  B  =  {
 g  |  E. z
 ( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred
 ( R ,  A ,  w ) ) ) ) ) }
 
Theoremfrrlem2 23637* Lemma for founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( g  e.  B  ->  Fun  g )
 
Theoremfrrlem3 23638* Lemma for founded recursion. An acceptable function's domain is a subset of  A. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( g  e.  B  ->  dom  g  C_  A )
 
Theoremfrrlem4 23639* Lemma for founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i  dom  h )  /\  A. a  e.  ( dom  g  i^i 
 dom  h ) ( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a
 )  =  ( a G ( ( g  |`  ( dom  g  i^i 
 dom  h ) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
 
Theoremfrrlem5 23640* Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremfrrlem5b 23641* Lemma for founded recursion. The union of a subclass of  B is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  Rel  U. C )
 
Theoremfrrlem5c 23642* Lemma for founded recursion. The union of a subclass of  B is a function. (Contributed by Paul Chapman, 29-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  Fun  U. C )
 
Theoremfrrlem5d 23643* Lemma for founded recursion. The domain of the union of a subset of  B is a subset of  A. (Contributed by Paul Chapman, 29-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  dom  U.  C  C_  A )
 
Theoremfrrlem5e 23644* Lemma for founded recursion. The domain of the union of a subset of  B is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  ( X  e.  dom  U.  C  ->  Pred ( R ,  A ,  X )  C_  dom  U.  C ) )
 
Theoremfrrlem6 23645* Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Rel  F
 
Theoremfrrlem7 23646* Lemma for founded recursion. The domain of  F is a subclass of  A. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 dom  F  C_  A
 
Theoremfrrlem10 23647* Lemma for founded recursion. The union of all acceptable functions is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Fun  F
 
Theoremfrrlem11 23648* Lemma for founded recursion. Here, we calculate the value of  F (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( y  e.  dom  F 
 ->  ( F `  y
 )  =  ( y G ( F  |`  Pred ( R ,  A ,  y ) ) ) )
 
16.7.22  Surreal Numbers
 
Syntaxcsur 23649 Declare the class of all surreal numbers (see df-no 23652).
 class  No
 
Syntaxcslt 23650 Declare the less than relationship over surreal numbers (see df-slt 23653).
 class  < s
 
Syntaxcbday 23651 Declare the birthday function for surreal numbers (see df-bday 23654).
 class  bday
 
Definitiondf-no 23652* Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Goshnor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 
1o and  2o, analagous to Goshnor's  (  -  ) and  (  +  ).

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in a effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

 |-  No  =  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
 
Definitiondf-slt 23653* Next, we introduce surreal less-than, a comparison relationship over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  < s  =  { <. f ,  g >.  |  (
 ( f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
 )  =  ( g `
  y )  /\  ( f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `
  x ) ) ) }
 
Definitiondf-bday 23654 Finally, we introduce the birthday function. This function maps each surreal to an ordinal. In our implementation, this is the domain of the sign function. The important properties of this function are established later. (Contributed by Scott Fenton, 11-Jun-2011.)
 |-  bday  =  ( x  e.  No  |->  dom  x )
 
Theoremelno 23655* Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
 |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
 
Theoremsltval 23656* The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
 <-> 
 E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `
  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `
  x ) ) ) )
 
Theorembdayval 23657 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  (
 bday `  A )  = 
 dom  A )
 
Theoremnofun 23658 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Fun 
 A )
 
Theoremnodmon 23659 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  dom 
 A  e.  On )
 
Theoremnorn 23660 The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  ran 
 A  C_  { 1o ,  2o } )
 
Theoremnodmord 23661 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Ord 
 dom  A )
 
Theoremelno2 23662 An alternative condition for membership in  No. (Contributed by Scott Fenton, 21-Mar-2012.)
 |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\ 
 ran  A  C_  { 1o ,  2o } ) )
 
Theoremelno3 23663 Another condition for membership in 
No. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  ( A  e.  No  <->  ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) )
 
Theoremsltval2 23664* Alternate expression for surreal less than. Two surreals obey surreal less than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
 <->  ( A `  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } ) ) )
 
Theoremnofv 23665 The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
 |-  ( A  e.  No  ->  ( ( A `  X )  =  (/)  \/  ( A `  X )  =  1o  \/  ( A `
  X )  =  2o ) )
 
Theoremnosgnn0 23666  (/) is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  (/) 
 e.  { 1o ,  2o }
 
Theoremnosgnn0i 23667 If  X is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  (/) 
 =/=  X
 
Theoremnoreson 23668 The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  On )  ->  ( A  |`  B )  e.  No )
 
Theoremsltsgn1 23669* If  A < s B, then the sign of  A at the first place they differ is either undefined or  1o (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( A `
  |^| { k  e. 
 On  |  ( A `
  k )  =/=  ( B `  k
 ) } )  =  (/)  \/  ( A `  |^|
 { k  e.  On  |  ( A `  k
 )  =/=  ( B `  k ) } )  =  1o ) ) )
 
Theoremsltsgn2 23670* If  A < s B, then the sign of  B at the first place they differ is either undefined or  2o (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( B `
  |^| { k  e. 
 On  |  ( A `
  k )  =/=  ( B `  k
 ) } )  =  (/)  \/  ( B `  |^|
 { k  e.  On  |  ( A `  k
 )  =/=  ( B `  k ) } )  =  2o ) ) )
 
Theoremsltintdifex 23671* If  A < s B, then the intersection of all the ordinals that have differing signs in  A and  B exists. (Contributed by Scott Fenton, 22-Feb-2012.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  _V ) )
 
Theoremsltres 23672 Lemma for axfe (future) . If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  X  e.  On )  ->  ( ( A  |`  X ) < s ( B  |`  X )  ->  A < s B ) )
 
Theoremnoxpsgn 23673 The cross product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  On  ->  ( A  X.  { X } )  e.  No )
 
Theoremnoxp1o 23674 The cross product of an ordinal and  { 1o } is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
 |-  ( A  e.  On  ->  ( A  X.  { 1o } )  e.  No )
 
Theoremnoxp2o 23675 The cross product of an ordinal and  { 2o } is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
 |-  ( A  e.  On  ->  ( A  X.  { 2o } )  e.  No )
 
16.7.23  Surreal Numbers: Ordering
 
Theoremaxsltsolem1 23676 Lemma for axsltso 23677. The sign expansion relationship totally orders the surreal signs. (Contributed by axsltsolem1, 8-Jun-2011.)
 |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/)
 } )
 
Theoremaxsltso 23677 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  < s  Or  No
 
Theoremsltirr 23678 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  -.  A < s A )
 
Theoremslttr 23679 Surreal less than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  C  e.  No )  ->  ( ( A <
 s B  /\  B < s C )  ->  A < s C ) )
 
Theoremsltasym 23680 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  -.  B < s A ) )
 
Theoremslttri 23681 Surreal less than obeys trichotomy. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  \/  A  =  B  \/  B < s A ) )
 
Theoremslttrieq2 23682 Trichotomy law for surreal less than. (Contributed by Scott Fenton, 22-Apr-2012.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( A  =  B  <->  ( -.  A < s B  /\  -.  B <
 s A ) ) )
 
16.7.24  Surreal Numbers: Birthday Function
 
Theoremaxbday 23683 The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
 |-  bday : No -onto-> On
 
Theorembdayfun 23684 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  Fun  bday
 
Theorembdayrn 23685 The birthday function's range is 
On (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ran  bday 
 =  On
 
Theorembdaydm 23686 The birthday function's domain is 
No (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  dom  bday 
 =  No
 
Theorembdayfn 23687 The birthday function is a function over  No (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  bday  Fn 
 No
 
Theorembdayelon 23688 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( bday `  A )  e. 
 On
 
Theoremnoprc 23689 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  No  e.  _V
 
16.7.25  Surreal Numbers: Density
 
Theoremaxdenselem1 23690 Lemma for axdense 23698. A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  A  Fn  ( bday `  A ) )
 
Theoremaxdenselem2 23691 Lemma for axfe (future) and axdense 23698. The value of a surreal at its birthday is  (/). (Shortened proof on 2012-Apr-14, SF) (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
 
Theoremaxdenselem3 23692* Lemma for axdense 23698. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No )  ->  ( ( bday `  A )  e.  ( bday `  B )  ->  E. a  e.  On  ( A `  a )  =/=  ( B `  a ) ) )
 
Theoremaxdenselem4 23693* Lemma for axdense 23698. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  A < s B )  ->  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  On )
 
Theoremaxdenselem5 23694* Lemma for axdense 23698. If the birthdays of two distinct surreals are equal, then the ordinal from axdenselem4 23693 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  |^| { a  e.  On  |  ( A `
  a )  =/=  ( B `  a
 ) }  e.  ( bday `  A ) )
 
Theoremaxdenselem6 23695* The restriction of a surreal to the abstraction from axdenselem4 23693 is still a surreal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  ( A  |` 
 |^| { a  e.  On  |  ( A `  a
 )  =/=  ( B `  a ) } )  e.  No )
 
Theoremaxdenselem7 23696* Lemma for axdense 23698. 
A and  B are equal at all elements of the abstraction. (Contributed by Scott Fenton, 17-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  ( C  e.  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) }  ->  ( A `  C )  =  ( B `  C ) ) )
 
Theoremaxdenselem8 23697* Lemma for axdense 23698. Give a condition for surreal less than when two surreals have the same birthday. (Contributed by Scott Fenton, 19-Jun-2011.)
 |-  (
 ( A  e.  No  /\  B  e.  No  /\  ( bday `  A )  =  ( bday `  B )
 )  ->  ( A < s B  <->  ( ( A `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } )  =  1o  /\  ( B `
  |^| { a  e. 
 On  |  ( A `
  a )  =/=  ( B `  a
 ) } )  =  2o ) ) )
 
Theoremaxdense 23698* Given two distinct surreals with the same birthday, there is an older surreal lying between the two of them. Alling's axiom (SD) (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  (
 ( ( A  e.  No  /\  B  e.  No )  /\  ( ( bday `  A )  =  (
 bday `  B )  /\  A < s B ) )  ->  E. x  e.  No  ( ( bday `  x )  e.  ( bday `  A )  /\  A < s x  /\  x < s B ) )
 
Theoremnocvxminlem 23699* Lemma for nocvxmin 23700. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  (
 ( A  C_  No  /\ 
 A. x  e.  A  A. y  e.  A  A. z  e.  No  (
 ( x < s
 z  /\  z < s y )  ->  z  e.  A ) )  ->  ( ( ( X  e.  A  /\  Y  e.  A )  /\  (
 ( bday `  X )  =  |^| ( bday " A )  /\  ( bday `  Y )  =  |^| ( bday " A ) ) ) 
 ->  -.  X < s Y ) )
 
Theoremnocvxmin 23700* Given a non-empty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  (
 ( A  =/=  (/)  /\  A  C_ 
 No  /\  A. x  e.  A  A. y  e.  A  A. z  e. 
 No  ( ( x < s z  /\  z < s y ) 
 ->  z  e.  A ) )  ->  E! w  e.  A  ( bday `  w )  =  |^| ( bday " A ) )
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