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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4bc3eq4 23501 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  (
 4  _C  3 )  =  4
 
Theorem4bc2eq6 23502 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 4  _C  2 )  =  6
 
Theoremhalfthird 23503 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  (
 ( 1  /  2
 )  -  ( 1 
 /  3 ) )  =  ( 1  / 
 6 )
 
Theorem5recm6rec 23504 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 ( 1  /  5
 )  -  ( 1 
 /  6 ) )  =  ( 1  / ; 3 0 )
 
18.7.6  Greatest common divisor and divisibility
 
Theorempdivsq 23505 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( P  e.  Prime  /\  M  e.  ZZ )  ->  ( P  ||  M  <->  P 
 ||  ( M ^
 2 ) ) )
 
Theoremdvdspw 23506 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N ) ) )
 
Theoremgcd32 23507 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( A 
 gcd  C )  gcd  B ) )
 
Theoremgcdabsorb 23508 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  B )  =  ( A  gcd  B ) )
 
18.7.7  Properties of relationships
 
Theorembrtp 23509 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
 
Theoremdftr6 23510 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
 
Theoremcoep 23511* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A (  _E  o.  R ) B  <->  E. x  e.  B  A R x )
 
Theoremcoepr 23512* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
 
Theoremdffr5 23513 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } )  C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
 
Theoremdfso2 23514 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  ( A  X.  A )  C_  ( R  u.  (  _I  u.  `' R ) ) ) )
 
Theoremdfpo2 23515 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Po  A  <->  ( ( R  i^i  (  _I  |`  A ) )  =  (/)  /\  (
 ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  R ) )
 
Theorembr8 23516* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  (
 g  =  G  ->  ( ze  <->  si ) )   &    |-  ( h  =  H  ->  (
 si 
 <->  rh ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  = 
 <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ph ) }   =>    |-  ( ( ( X  e.  S  /\  A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q  /\  E  e.  Q )  /\  ( F  e.  Q  /\  G  e.  Q  /\  H  e.  Q )
 )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  rh ) )
 
Theorembr6 23517* Substitution for an six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ph ) }   =>    |-  (
 ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q  /\  C  e.  Q )  /\  ( D  e.  Q  /\  E  e.  Q  /\  F  e.  Q )
 )  ->  ( <. A ,  <. B ,  C >.
 >. R <. D ,  <. E ,  F >. >.  <->  ze ) )
 
Theorembr4 23518* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  = 
 <. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) }   =>    |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q ) )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
 
Theoremdfres3 23519 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A )
 )
 
Theoremcnvco1 23520 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 23521 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
18.7.8  Properties of functions and mappings
 
Theoremfunpsstri 23522 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  (
 ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F 
 C.  G  \/  F  =  G  \/  G  C.  F ) )
 
Theoremfundmpss 23523 If a class  F is a proper subset of a function  G, then  dom  F  C.  dom  G. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( Fun  G  ->  ( F  C.  G  ->  dom  F  C.  dom 
 G ) )
 
Theoremfvresval 23524 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( ( F  |`  B ) `
  A )  =  ( F `  A )  \/  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremmptrel 23525 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 23526 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( Fun  F  /\  Fun 
 G  /\  dom  F  =  dom  G )  ->  ( F  =  G  <->  F  C_  G ) )
 
Theoremfununiq 23527 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( Fun  F  ->  ( ( A F B  /\  A F C ) 
 ->  B  =  C ) )
 
Theoremfunbreq 23528 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( ( Fun  F  /\  A F B ) 
 ->  ( A F C  <->  B  =  C ) )
 
Theoremmpteq12d 23529 An equality inference for the maps to notation. Compare mpteq12dv 4099. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremfprb 23530* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  =/=  B  ->  ( F : { A ,  B } --> R  <->  E. x  e.  R  E. y  e.  R  F  =  { <. A ,  x >. ,  <. B ,  y >. } ) )
 
Theorembr1steq 23531 Uniqueness condition for binary relationship over the  1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 1st C  <->  C  =  A )
 
Theorembr2ndeq 23532 Uniqueness condition for binary relationship over the  2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 2nd C  <->  C  =  B )
 
Theoremdfdm5 23533 Definition of domain in terms of 
1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A )
 
Theoremdfrn5 23534 Definition of range in terms of 
2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A )
 
18.7.9  Epsilon induction
 
Theoremsetinds 23535* Principle of  _E induction (set induction). If a property passes from all elements of  x to  x itself, then it holds for all  x. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( A. y  e.  x  [. y  /  x ]. ph 
 ->  ph )   =>    |-  ph
 
Theoremsetinds2f 23536*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
Theoremsetinds2 23537*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
18.7.10  Ordinal numbers
 
Theoremelpotr 23538* A class of transitive sets is partially ordered by  _E. (Contributed by Scott Fenton, 15-Oct-2010.)
 |-  ( A. z  e.  A  Tr  z  ->  _E  Po  A )
 
Theoremdford5reg 23539 Given ax-reg 7301, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
 |-  ( Ord  A  <->  ( Tr  A  /\  _E  Or  A ) )
 
Theoremdfon2lem1 23540 Lemma for dfon2 23549. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  Tr  U.
 { x  |  (
 ph  /\  Tr  x  /\  ps ) }
 
Theoremdfon2lem2 23541* Lemma for dfon2 23549 (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A
 
Theoremdfon2lem3 23542* Lemma for dfon2 23549. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  ->  ( Tr  A  /\  A. z  e.  A  -.  z  e.  z ) ) )
 
Theoremdfon2lem4 23543* Lemma for dfon2 23549. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  /\  A. y ( ( y 
 C.  B  /\  Tr  y )  ->  y  e.  B ) )  ->  ( A  C_  B  \/  B  C_  A ) )
 
Theoremdfon2lem5 23544* Lemma for dfon2 23549. Two sets satisfying the new definition also satisfy trichotomy with respect to 
e. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  /\  A. y ( ( y 
 C.  B  /\  Tr  y )  ->  y  e.  B ) )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
 )
 
Theoremdfon2lem6 23545* Lemma for dfon2 23549. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  (
 ( Tr  S  /\  A. x  e.  S  A. z ( ( z 
 C.  x  /\  Tr  z )  ->  z  e.  x ) )  ->  A. y ( ( y 
 C.  S  /\  Tr  y )  ->  y  e.  S ) )
 
Theoremdfon2lem7 23546* Lemma for dfon2 23549. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   =>    |-  ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  ->  ( B  e.  A  ->  A. y ( ( y  C.  B  /\  Tr  y )  ->  y  e.  B ) ) )
 
Theoremdfon2lem8 23547* Lemma for dfon2 23549. The intersection of a non-empty class  A of new ordinals is itself a new ordinal and is contained within  A (Contributed by Scott Fenton, 26-Feb-2011.)
 |-  (
 ( A  =/=  (/)  /\  A. x  e.  A  A. y
 ( ( y  C.  x  /\  Tr  y ) 
 ->  y  e.  x ) )  ->  ( A. z ( ( z 
 C.  |^| A  /\  Tr  z )  ->  z  e. 
 |^| A )  /\  |^|
 A  e.  A ) )
 
Theoremdfon2lem9 23548* Lemma for dfon2 23549. A class of new ordinals is well-founded by  _E. (Contributed by Scott Fenton, 3-Mar-2011.)
 |-  ( A. x  e.  A  A. y ( ( y 
 C.  x  /\  Tr  y )  ->  y  e.  x )  ->  _E  Fr  A )
 
Theoremdfon2 23549*  On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)
 |-  On  =  { x  |  A. y ( ( y 
 C.  x  /\  Tr  y )  ->  y  e.  x ) }
 
Theoremdomep 23550 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
 |-  dom  _E  =  _V
 
Theoremrdgprc0 23551 The value of the recursive definition generator at  (/) when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
 
Theoremrdgprc 23552 The value of the recursive definition generator when  I is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( -.  I  e.  _V  ->  rec ( F ,  I )  =  rec ( F ,  (/) ) )
 
Theoremdfrdg2 23553* Alternate definition of the recursive function generator when  I is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( I  e.  V  ->  rec ( F ,  I
 )  =  U. {
 f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  if ( y  =  (/) ,  I ,  if ( Lim  y ,  U. ( f "
 y ) ,  ( F `  ( f `  U. y ) ) ) ) ) } )
 
Theoremdfrdg3 23554* Generalization of dfrdg2 23553 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  rec ( F ,  I )  =  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  if (
 y  =  (/) ,  if ( I  e.  _V ,  I ,  (/) ) ,  if ( Lim  y ,  U. ( f "
 y ) ,  ( F `  ( f `  U. y ) ) ) ) ) }
 
18.7.11  Defined equality axioms
 
Theoremaxextdfeq 23555 A version of ax-ext 2265 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  E. z
 ( ( z  e.  x  ->  z  e.  y )  ->  ( ( z  e.  y  ->  z  e.  x )  ->  ( x  e.  w  ->  y  e.  w ) ) )
 
Theoremax13dfeq 23556 A version of ax-13 1690 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  E. z
 ( ( z  e.  x  ->  z  e.  y )  ->  ( w  e.  x  ->  w  e.  y ) )
 
Theoremaxextdist 23557 ax-ext 2265 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( -.  A. z  z  =  x  /\  -. 
 A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y ) )
 
Theoremaxext4dist 23558 axext4 2268 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( -.  A. z  z  =  x  /\  -. 
 A. z  z  =  y )  ->  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y
 ) ) )
 
Theorem19.12b 23559* 19.12vv 1850 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x A. y (
 ph  ->  ps )  <->  A. y E. x ( ph  ->  ps )
 )
 
Theoremexnel 23560 There is always a set not in  y. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  E. x  -.  x  e.  y
 
Theoremdistel 23561 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4191 and elirrv 7306.) (Contributed by Scott Fenton, 15-Dec-2010.)
 |-  ( -.  A. y  y  =  x  <->  -.  A. y  -.  x  e.  y )
 
Theoremaxextndbi 23562 axextnd 8208 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
 |-  E. z
 ( x  =  y  <-> 
 ( z  e.  x  <->  z  e.  y ) )
 
18.7.12  Hypothesis builders
 
Theoremhbntg 23563 A more general form of hbnt 1728. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. x ps )  ->  ( -.  ps  ->  A. x  -.  ph ) )
 
Theoremhbimtg 23564 A more general and closed form of hbim 1729. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
 ( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
 
Theoremhbaltg 23565 A more general and closed form of hbal 1714. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. y ps )  ->  ( A. x ph  ->  A. y A. x ps ) )
 
Theoremhbng 23566 A more general form of hbn 1724. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( -.  ps  ->  A. x  -.  ph )
 
Theoremhbimg 23567 A more general form of hbim 1729. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   &    |-  ( ch  ->  A. x th )   =>    |-  (
 ( ps  ->  ch )  ->  A. x ( ph  ->  th ) )
 
18.7.13  The Predecessor Class
 
Syntaxcpred 23568 The predecessors symbol.
 class  Pred ( R ,  A ,  X )
 
Definitiondf-pred 23569 Define the predecessor class of a relationship. This is the class of all elements  y of  A such that  y R X (see elpred 23578) . (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } )
 )
 
Theorempredeq1 23570 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  =  S  ->  Pred
 ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
 
Theorempredeq2 23571 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  Pred
 ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
 
Theorempredeq3 23572 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  Pred
 ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
 
Theorempredpredss 23573 If  A is a subset of  B, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  C_  B  ->  Pred ( R ,  A ,  X )  C_  Pred ( R ,  B ,  X ) )
 
Theorempredss 23574 The predecessor class of  A is a subset of  A (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  Pred ( R ,  A ,  X )  C_  A
 
Theoremsspred 23575 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)
 |-  (
 ( B  C_  A  /\  Pred ( R ,  A ,  X )  C_  B )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
 
Theoremdfpred2 23576* An alternate definition of predecessor class when  X is a set. (Contributed by Scott Fenton, 8-Feb-2011.)
 |-  X  e.  _V   =>    |- 
 Pred ( R ,  A ,  X )  =  ( A  i^i  {
 y  |  y R X } )
 
Theoremelpredim 23577 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
 |-  X  e.  _V   =>    |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X )
 
Theoremelpred 23578 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
 |-  Y  e.  _V   =>    |-  ( X  e.  D  ->  ( Y  e.  Pred ( R ,  A ,  X )  <->  ( Y  e.  A  /\  Y R X ) ) )
 
Theoremelpredg 23579 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
 |-  (
 ( X  e.  B  /\  Y  e.  A ) 
 ->  ( Y  e.  Pred ( R ,  A ,  X )  <->  Y R X ) )
 
Theorempredreseq 23580* Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)
 |-  X  e.  _V   =>    |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( F  |`  Pred ( R ,  A ,  X )
 )  =  ( G  |`  Pred ( R ,  A ,  X )
 ) 
 <-> 
 A. y  e.  A  ( y R X  ->  ( F `  y
 )  =  ( G `
  y ) ) ) )
 
Theorempredasetex 23581 The predecessor class exists when 
A does. (Contributed by Scott Fenton, 8-Feb-2011.)
 |-  A  e.  _V   =>    |- 
 Pred ( R ,  A ,  X )  e.  _V
 
Theoremcbvsetlike 23582* Change the bound variable in the statement stating that  R is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V  <->  A. y  e.  A  Pred ( R ,  A ,  y )  e.  _V )
 
Theoremdffr4 23583* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  Fr  A  <->  A. x ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  Pred ( R ,  x ,  y
 )  =  (/) ) )
 
Theorempredel 23584 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
 
Theorempredpo 23585 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  (
 ( R  Po  A  /\  X  e.  A ) 
 ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Pred ( R ,  A ,  Y )  C_  Pred ( R ,  A ,  X ) ) )
 
Theorempredso 23586 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 ( R  Or  A  /\  X  e.  A ) 
 ->  ( Y  e.  Pred ( R ,  A ,  X )  ->  Pred ( R ,  A ,  Y )  C_  Pred ( R ,  A ,  X ) ) )
 
Theorempredbrg 23587 Closed form of elpredim 23577. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
 |-  (
 ( X  e.  V  /\  Y  e.  Pred ( R ,  A ,  X ) )  ->  Y R X )
 
Theoremsetlikespec 23588 If  R is set-like in  A then all predecessors classes of elements of  A exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  e.  _V )
 
Theorempredidm 23589 Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)
 |-  Pred ( R ,  Pred ( R ,  A ,  X ) ,  X )  =  Pred ( R ,  A ,  X )
 
Theorempredin 23590 Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
 |-  Pred ( R ,  ( A  i^i  B ) ,  X )  =  (
 Pred ( R ,  A ,  X )  i^i  Pred ( R ,  B ,  X )
 )
 
Theorempredun 23591 Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
 |-  Pred ( R ,  ( A  u.  B ) ,  X )  =  (
 Pred ( R ,  A ,  X )  u.  Pred ( R ,  B ,  X )
 )
 
Theorempreddif 23592 Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)
 |-  Pred ( R ,  ( A 
 \  B ) ,  X )  =  (
 Pred ( R ,  A ,  X )  \ 
 Pred ( R ,  B ,  X )
 )
 
Theorempredep 23593 The predecessor under the epsilon relationship is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( X  e.  B  ->  Pred
 (  _E  ,  A ,  X )  =  ( A  i^i  X ) )
 
Theorempredon 23594 For an ordinal, the predecessor under  _E and  On is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)
 |-  ( A  e.  On  ->  Pred
 (  _E  ,  On ,  A )  =  A )
 
Theoremepsetlike 23595 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)
 |-  A. x  e.  A  Pred (  _E  ,  A ,  x )  e.  _V
 
Theoremsetlikess 23596* If  R is set-like over  A, then it is set-like over any subclass of  A. (Contributed by Scott Fenton, 28-Mar-2011.)
 |-  (
 ( A  C_  B  /\  A. x  e.  B  Pred ( R ,  B ,  x )  e.  _V )  ->  A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V )
 
Theorempreddowncl 23597* A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)
 |-  (
 ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B )  ->  ( X  e.  B  ->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X )
 ) )
 
Theorempredpoirr 23598 Given a partial ordering,  X is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
 |-  ( R  Po  A  ->  -.  X  e.  Pred ( R ,  A ,  X )
 )
 
Theorempredfrirr 23599 Given a well-founded relationship, 
X is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
 |-  ( R  Fr  A  ->  -.  X  e.  Pred ( R ,  A ,  X )
 )
 
Theorempred0 23600 The predecessor class over 
(/) is always 
(/) (Contributed by Scott Fenton, 16-Apr-2011.)
 |-  Pred ( R ,  (/) ,  X )  =  (/)
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