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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-bday 24301 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 24302* 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 24303* 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 24304 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( A  e.  No  ->  (
 bday `  A )  = 
 dom  A )
 
Theoremnofun 24305 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Fun 
 A )
 
Theoremnodmon 24306 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  dom 
 A  e.  On )
 
Theoremnorn 24307 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 } )
 
Theoremnofnbday 24308 A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  A  Fn  ( bday `  A ) )
 
Theoremnodmord 24309 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  Ord 
 dom  A )
 
Theoremelno2 24310 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 24311 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 24312* 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 24313 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 24314  (/) is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  (/) 
 e.  { 1o ,  2o }
 
Theoremnosgnn0i 24315 If  X is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  (/) 
 =/=  X
 
Theoremnoreson 24316 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 24317* 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 24318* 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 24319* 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 24320 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 24321 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 24322 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 24323 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 )
 
18.7.23  Surreal Numbers: Ordering
 
Theoremsltsolem1 24324 Lemma for sltso 24325. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  Or  ( { 1o ,  2o }  u.  { (/)
 } )
 
Theoremsltso 24325 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  < s  Or  No
 
Theoremsltirr 24326 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  ( A  e.  No  ->  -.  A < s A )
 
Theoremslttr 24327 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 24328 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 24329 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 24330 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 ) ) )
 
18.7.24  Surreal Numbers: Birthday Function
 
Theorembdayfo 24331 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 24332 The birthday function is a function. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  Fun  bday
 
Theorembdayrn 24333 The birthday function's range is 
On (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ran  bday 
 =  On
 
Theorembdaydm 24334 The birthday function's domain is 
No (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  dom  bday 
 =  No
 
Theorembdayfn 24335 The birthday function is a function over  No (Contributed by Scott Fenton, 30-Jun-2011.)
 |-  bday  Fn 
 No
 
Theorembdayelon 24336 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
 |-  ( bday `  A )  e. 
 On
 
Theoremnoprc 24337 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
 |-  -.  No  e.  _V
 
18.7.25  Surreal Numbers: Density
 
Theoremfvnobday 24338 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 ) )  =  (/) )
 
18.7.26  Surreal Numbers: Density
 
Theoremnodenselem3 24339* Lemma for nodense 24345. 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 ) ) )
 
Theoremnodenselem4 24340* Lemma for nodense 24345. 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 )
 
Theoremnodenselem5 24341* Lemma for nodense 24345. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 24340 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 ) )
 
Theoremnodenselem6 24342* The restriction of a surreal to the abstraction from nodenselem4 24340 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 )
 
Theoremnodenselem7 24343* Lemma for nodense 24345. 
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 ) ) )
 
Theoremnodenselem8 24344* Lemma for nodense 24345. 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 ) ) )
 
Theoremnodense 24345* 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 24346* Lemma for nocvxmin 24347. 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 24347* 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 ) )
 
18.7.27  Surreal Numbers: Upper and Lower Bounds
 
Theoremnobndlem1 24348 Lemma for nobndup 24356 and nobnddown 24357. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.)
 |-  ( A  e.  V  ->  suc  U. ( bday " A )  e. 
 On )
 
Theoremnobndlem2 24349* Lemma for nobndup 24356 and nobnddown 24357. Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  C  e.  On )
 
Theoremnobndlem3 24350* Lemma for nobndup 24356 and nobnddown 24357. Calculate the birthday of  ( C  X.  { X } ). (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  ( bday `  ( C  X.  { X } )
 )  =  C )
 
Theoremnobndlem4 24351* Lemma for nobndup 24356 and nobnddown 24357. The infimum of the class of all ordinals such that  A is not  X is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  No  -> 
 |^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  On )
 
Theoremnobndlem5 24352* Lemma for nobndup 24356 and nobnddown 24357. There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   =>    |-  ( A  e.  No  ->  ( A `  |^| { x  e.  On  |  ( A `
  x )  =/= 
 X } )  =/= 
 X )
 
Theoremnobndlem6 24353* Lemma for nobndup 24356 and nobnddown 24357. Given an element  A of  F, then the first position where it differs from  X is strictly less than  C (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  A  e.  F ) 
 ->  |^| { x  e. 
 On  |  ( A `
  x )  =/= 
 X }  e.  C )
 
Theoremnobndlem7 24354* Lemma for nobndup 24356 and nobnddown 24357. Calculate the value of  ( C  X.  { X } ) at the minimal ordinal whose value is different from  X. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  X  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  X }   =>    |-  (
 ( F  C_  No  /\  A  e.  F ) 
 ->  ( ( C  X.  { X } ) `  |^|
 { x  e.  On  |  ( A `  x )  =/=  X } )  =  X )
 
Theoremnobndlem8 24355* Lemma for nobndup 24356 and nobnddown 24357. Bound the birthday of  ( C  X.  { S } ) above. (Contributed by Scott Fenton, 10-Apr-2017.)
 |-  S  e.  { 1o ,  2o }   &    |-  C  =  |^| { a  e.  On  |  A. n  e.  F  E. b  e.  a  ( n `  b )  =/=  S }   =>    |-  (
 ( F  C_  No  /\  F  e.  A ) 
 ->  ( bday `  ( C  X.  { S } )
 )  C_  suc  U. ( bday " F ) )
 
Theoremnobndup 24356* Any set of surreals is bounded above by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
 |-  (
 ( A  C_  No  /\  A  e.  V ) 
 ->  E. x  e.  No  ( A. y  e.  A  y < s x  /\  ( bday `  x )  C_ 
 suc  U. ( bday " A ) ) )
 
Theoremnobnddown 24357* Any set of surreals is bounded below by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
 |-  (
 ( A  C_  No  /\  A  e.  V ) 
 ->  E. x  e.  No  ( A. y  e.  A  x < s y  /\  ( bday `  x )  C_ 
 suc  U. ( bday " A ) ) )
 
18.7.28  Surreal Numbers: Full-Eta Property
 
Theoremnofulllem1 24358* Lemma for nofull (future) . The full statement of the axiom when  R is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  ( R  =  (/)  ->  (
 ( ( L  C_  No  /\  L  e.  V )  /\  ( R  C_  No  /\  R  e.  W )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  E. z  e.  No  ( A. x  e.  L  x < s
 z  /\  A. y  e.  R  z < s
 y  /\  ( bday `  z )  C_  suc  U. ( bday " ( L  u.  R ) ) ) ) )
 
Theoremnofulllem2 24359* Lemma for nofull (future) . The full statement of the axiom when  L is empty. (Contributed by Scott Fenton, 3-Aug-2011.)
 |-  ( L  =  (/)  ->  (
 ( ( L  C_  No  /\  L  e.  V )  /\  ( R  C_  No  /\  R  e.  W )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  E. z  e.  No  ( A. x  e.  L  x < s
 z  /\  A. y  e.  R  z < s
 y  /\  ( bday `  z )  C_  suc  U. ( bday " ( L  u.  R ) ) ) ) )
 
Theoremnofulllem3 24360 Lemma for nofull (future) . Restriction of surreal number to a superset of its birthday does not change anything. (Contributed by Scott Fenton, 25-Apr-2017.)
 |-  (
 ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  ( X  |`  U. ( bday " S ) )  =  X )
 
Theoremnofulllem4 24361* Lemma for nofull (future) . Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 25-Apr-2017.)
 |-  M  =  |^| { a  e. 
 On  |  A. x  e.  L  A. y  e.  R  ( x  |`  a )  =/=  (
 y  |`  a ) }   =>    |-  (
 ( ( L  C_  No  /\  L  e.  V )  /\  ( R  C_  No  /\  R  e.  W )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  M  e.  On )
 
Theoremnofulllem5 24362* Lemma for nofull (future) . Here, we introduce a new surreal number  X. Eventually, we will show that either  X or a related surreal number has the required properties for the final theorem. We begin by calculating the domain of  X. (Contributed by Scott Fenton, 1-May-2017.)
 |-  M  =  |^| { a  e. 
 On  |  A. x  e.  L  A. y  e.  R  ( x  |`  a )  =/=  (
 y  |`  a ) }   &    |-  S  =  { f  |  E. g  e.  L  E. h  e.  R  E. a  e.  M  ( ( g  |`  a )  =  f 
 /\  ( h  |`  a )  =  f
 ) }   &    |-  X  =  U. S   =>    |-  ( ( ( L 
 C_  No  /\  L  e.  V )  /\  ( R 
 C_  No  /\  R  e.  W )  /\  A. x  e.  L  A. y  e.  R  x < s
 y )  ->  dom  X  =  U. M )
 
18.7.29  Symmetric difference
 
Syntaxcsymdif 24363 Declare the syntax for symmetric difference.
 class  ( A(++)
 B )
 
Definitiondf-symdif 24364 Define the symmetric difference of two classes. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A(++) B )  =  ( ( A  \  B )  u.  ( B  \  A ) )
 
Theoremsymdifcom 24365 Symmetric difference commutes. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) B )  =  ( B(++) A )
 
Theoremsymdifeq1 24366 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A  =  B  ->  ( A(++) C )  =  ( B(++) C ) )
 
Theoremsymdifeq2 24367 Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A  =  B  ->  ( C(++) A )  =  ( C(++) B ) )
 
Theoremnfsymdif 24368 Hypothesis builder for symmetric difference. (Contributed by Scott Fenton, 19-Feb-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A(++) B )
 
Theoremelsymdif 24369 Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A  e.  ( B(++) C )  <->  -.  ( A  e.  B 
 <->  A  e.  C ) )
 
Theoremsymdif0 24370 Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) (/) )  =  A
 
TheoremsymdifV 24371 Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) _V )  =  ( _V  \  A )
 
Theoremsymdifid 24372 Symmetric difference yields the empty class with the same argument twice. (Contributed by Scott Fenton, 25-Apr-2012.)
 |-  ( A(++) A )  =  (/)
 
Theoremsymdifass 24373 Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)
 |-  ( A(++) ( B(++) C ) )  =  ( ( A(++) B )(++) C )
 
Theorembrsymdif 24374 The binary relationship of a symmetric difference. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  ( A ( R(++) S ) B  <->  -.  ( A R B 
 <->  A S B ) )
 
18.7.30  Quantifier-free definitions
 
Syntaxctxp 24375 Declare the syntax for tail cross product.
 class  ( A 
 (x)  B )
 
Syntaxcpprod 24376 Declare the syntax for the parallel product.
 class pprod ( R ,  S )
 
Syntaxcsset 24377 Declare the subset relationship class.
 class  SSet
 
Syntaxctrans 24378 Declare the transitive set class.
 class  Trans
 
Syntaxcbigcup 24379 Declare the set union relationship.
 class  Bigcup
 
Syntaxcfix 24380 Declare the syntax for the fixpoints of a class.
 class  Fix A
 
Syntaxclimits 24381 Declare the class of limit ordinals.
 class  Limits
 
Syntaxcfuns 24382 Declare the syntax for the class of all function.
 class  Funs
 
Syntaxcsingle 24383 Declare the syntax for the singleton function.
 class Singleton
 
Syntaxcsingles 24384 Declare the syntax for the class of all singletons.
 class  Singletons
 
Syntaxcimage 24385 Declare the syntax for the image functor.
 class Image A
 
Syntaxccart 24386 Declare the syntax for the cartesian function.
 class Cart
 
Syntaxcimg 24387 Declare the syntax for the image function.
 class Img
 
Syntaxcdomain 24388 Declare the syntax for the domain function.
 class Domain
 
Syntaxcrange 24389 Declare the syntax for the range function.
 class Range
 
Syntaxcapply 24390 Declare the syntax for the application function.
 class Apply
 
Syntaxccup 24391 Declare the syntax for the cup function.
 class Cup
 
Syntaxccap 24392 Declare the syntax for the cap function.
 class Cap
 
Syntaxcsuccf 24393 Declare the syntax for the successor function.
 class Succ
 
Syntaxcfunpart 24394 Declare the syntax for the functional part functor.
 class Funpart F
 
Syntaxcfullfn 24395 Declare the syntax for the full function functor.
 class FullFun F
 
Syntaxcrestrict 24396 Declare the syntax for the restriction function.
 class Restrict
 
Definitiondf-txp 24397 Define the tail cross of two classes. Membership in this class is defined by txpss3v 24420 and brtxp 24422. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  ( A  (x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
 
Definitiondf-pprod 24398 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 24426 and brpprod 24427. (Contributed by Scott Fenton, 11-Apr-2014.)
 |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
 
Definitiondf-sset 24399 Define the subset class. For the value, see brsset 24431. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
 
Definitiondf-trans 24400 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Trans  =  ( _V  \  ran  (
 (  _E  o.  _E  )  \  _E  ) )
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