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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtrpredtr 23401 The transitive predecessors are transitive in  R and 
A (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  ( Y  e.  TrPred ( R ,  A ,  X )  ->  Pred ( R ,  A ,  Y )  C_  TrPred ( R ,  A ,  X ) ) )
 
Theoremtrpredmintr 23402* The transitive predecessors form the smallest class transitive in  R and  A. That is, if  B is another  R,  A transitive class containing  Pred ( R ,  A ,  X ), then  TrPred ( R ,  A ,  X )  C_  B (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( X  e.  A  /\  R Se  A ) 
 /\  ( A. y  e.  B  Pred ( R ,  A ,  y )  C_  B  /\  Pred ( R ,  A ,  X )  C_  B ) )  ->  TrPred ( R ,  A ,  X )  C_  B )
 
Theoremtrpredelss 23403 Given a transitive predecessor  Y of  X, the transitive predecessors of  Y are a subset of the transitive predecessors of  X. (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  ( Y  e.  TrPred ( R ,  A ,  X )  ->  TrPred ( R ,  A ,  Y )  C_  TrPred ( R ,  A ,  X ) ) )
 
Theoremdftrpred3g 23404* The transitive predecessors of  X are equal to the predecessors of  X together with their transitive predecessors. (Contributed by Scott Fenton, 26-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  TrPred ( R ,  A ,  X )  =  ( Pred ( R ,  A ,  X )  u.  U_ y  e.  Pred  ( R ,  A ,  X ) TrPred ( R ,  A ,  y )
 ) )
 
Theoremdftrpred4g 23405* Another recursive expression for the transitive predecessors. (Contributed by Scott Fenton, 27-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  TrPred ( R ,  A ,  X )  =  U_ y  e.  Pred  ( R ,  A ,  X )
 ( { y }  u.  TrPred ( R ,  A ,  y )
 ) )
 
Theoremtrpredpo 23406 If  R partially orders  A, then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( R  Po  A  /\  X  e.  A  /\  R Se  A )  ->  TrPred ( R ,  A ,  X )  =  Pred ( R ,  A ,  X ) )
 
Theoremtrpred0 23407 The class of transitive predecessors is empty when  A is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
 |-  TrPred ( R ,  (/) ,  X )  =  (/)
 
Theoremtrpredex 23408 The transitive predecessors of a relation form a set (NOTE: this is the first theorem in the transitive predecessor series that requires infinity). (Contributed by Scott Fenton, 18-Feb-2011.)
 |-  TrPred ( R ,  A ,  X )  e.  _V
 
Theoremtrpredrec 23409* If  Y is an  R,  A transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between  Y and  X (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  ( Y  e.  TrPred ( R ,  A ,  X )  ->  ( Y  e.  Pred
 ( R ,  A ,  X )  \/  E. z  e.  TrPred  ( R ,  A ,  X ) Y R z ) ) )
 
16.6.17  Founded Induction
 
Theoremfrmin 23410* Every (possibly proper) subclass of a class  A with a founded, set-like relation  R has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 23373 and tz7.5 4306. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  Fr  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  B  =/=  (/) ) ) 
 ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
 
Theoremfrind 23411* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 23410). This principle states that if  B is a subclass of a founded class  A with the property that every element of  B whose initial segment is included in  A is is itself equal to  A. Compare wfi 23375 and tfi 4535, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  Fr  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
 
Theoremfrindi 23412* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 23410). This principle states that if  B is a subclass of a founded class  A with the property that every element of  B whose initial segment is included in  A is is itself equal to  A. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   =>    |-  ( ( B 
 C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  =  B )
 
Theoremfrinsg 23413* 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, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( ( R  Fr  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremfrins 23414* 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 23415* 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 23416* 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 23417* 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 23418* 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 23419* 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 )
 
16.6.18  Ordering Ordinal Sequences
 
Theoremorderseqlem 23420* Lemma for poseq 23421 and soseq 23422. 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 23421* 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 23422* 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
 
16.6.19  Well-founded recursion
 
Theoremwfr3g 23423* 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 23424* 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 23425* 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 23426* 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 23427* 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 23428* 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 23429* 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 23430* 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 23431* 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 23432* 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 23433* 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 23434* 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 23435* 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 23436* Lemma for well-founded recursion. From here through wfrlem16 23439, 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 23437* 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 23438* 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 23439* 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 23430),  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 23440* 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 23441* 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 23442* Generalize wfr2 23441 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 23443* 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 23440 and wfr2 23441 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 )
 
16.6.20  Transfinite recursion via Well-founded recursion
 
TheoremtfrALTlem 23444* 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 23445* tfr1 6299 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 23446* tfr2 6300 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 23447* tfr3 6301 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 )
 
16.6.21  Founded Recursion
 
Theoremfrr3g 23448* 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 23449* 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 23450* 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 23451* 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 23452* 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 23453* 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 23454* 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 23455* 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 23456* 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 23457* 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 23458* 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 23459* 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 23460* 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 23461* 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 ) ) ) )
 
16.6.22  Surreal Numbers
 
Syntaxcsur 23462 Declare the class of all surreal numbers (see df-no 23465).
 class  No
 
Syntaxcslt 23463 Declare the less than relationship over surreal numbers (see df-slt 23466).
 class  < s
 
Syntaxcbday 23464 Declare the birthday function for surreal numbers (see df-bday 23467).
 class  bday
 
Definitiondf-no 23465* 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 23466* 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 23467 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 23468* 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 23469* 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 23470 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  (
 bday `  A )  = 
 dom  A )
 
Theoremnofun 23471 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Fun 
 A )
 
Theoremnodmon 23472 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  dom 
 A  e.  On )
 
Theoremnorn 23473 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 23474 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Ord 
 dom  A )
 
Theoremelno2 23475 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 23476 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 23477* 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 23478 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 23479  (/) is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  (/) 
 e.  { 1o ,  2o }
 
Theoremnosgnn0i 23480 If  X is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  (/) 
 =/=  X
 
Theoremnoreson 23481 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 23482* 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 23483* 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 23484* 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 23485 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 23486 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 23487 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 23488 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 )
 
16.6.23  Surreal Numbers: Ordering
 
Theoremaxsltsolem1 23489 Lemma for axsltso 23490. The sign expansion relationship totally orders the surreal signs. (Contributed by axsltsolem1, 8-Jun-2011.)
 |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/)
 } )
 
Theoremaxsltso 23490 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  < s  Or  No
 
Theoremsltirr 23491 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  -.  A < s A )
 
Theoremslttr 23492 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 23493 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 23494 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 23495 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 ) ) )
 
16.6.24  Surreal Numbers: Birthday Function
 
Theoremaxbday 23496 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 23497 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  Fun  bday
 
Theorembdayrn 23498 The birthday function's range is 
On (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ran  bday 
 =  On
 
Theorembdaydm 23499 The birthday function's domain is 
No (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  dom  bday 
 =  No
 
Theorembdayfn 23500 The birthday function is a function over  No (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  bday  Fn 
 No
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