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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfon2lem8 23501* Lemma for dfon2 23503. 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 23502* Lemma for dfon2 23503. 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 23503*  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 23504 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
 |-  dom  _E  =  _V
 
Theoremrdgprc0 23505 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 23506 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 23507* 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 23508* Generalization of dfrdg2 23507 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 ) ) ) ) ) }
 
16.7.11  Defined equality axioms
 
Theoremaxextdfeq 23509 A version of ax-ext 2237 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 23510 A version of ax-13 1625 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 23511 ax-ext 2237 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 23512 axext4 2240 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 23513* 19.12vv 2032 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 23514 There is always a set not in  y. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  E. x  -.  x  e.  y
 
Theoremdistel 23515 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4150 and elirrv 7265.) (Contributed by Scott Fenton, 15-Dec-2010.)
 |-  ( -.  A. y  y  =  x  <->  -.  A. y  -.  x  e.  y )
 
Theoremaxextndbi 23516 axextnd 8167 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
 |-  E. z
 ( x  =  y  <-> 
 ( z  e.  x  <->  z  e.  y ) )
 
16.7.12  Hypothesis builders
 
Theoremhbntg 23517 A more general form of hbnt 1717. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. x ps )  ->  ( -.  ps  ->  A. x  -.  ph ) )
 
Theoremhbimtg 23518 A more general and closed form of hbim 1723. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
 ( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
 
Theoremhbaltg 23519 A more general and closed form of hbal 1567. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. y ps )  ->  ( A. x ph  ->  A. y A. x ps ) )
 
Theoremhbng 23520 A more general form of hbn 1722. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( -.  ps  ->  A. x  -.  ph )
 
Theoremhbimg 23521 A more general form of hbim 1723. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   &    |-  ( ch  ->  A. x th )   =>    |-  (
 ( ps  ->  ch )  ->  A. x ( ph  ->  th ) )
 
16.7.13  The Predecessor Class
 
Syntaxcpred 23522 The predecessors symbol.
 class  Pred ( R ,  A ,  X )
 
Definitiondf-pred 23523 Define the predecessor class of a relationship. This is the class of all elements  y of  A such that  y R X (see elpred 23532) . (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } )
 )
 
Theorempredeq1 23524 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  =  S  ->  Pred
 ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
 
Theorempredeq2 23525 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  Pred
 ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
 
Theorempredeq3 23526 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  Pred
 ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
 
Theorempredpredss 23527 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 23528 The predecessor class of  A is a subset of  A (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  Pred ( R ,  A ,  X )  C_  A
 
Theoremsspred 23529 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 23530* 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 23531 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 23532 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 23533 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 23534* 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 23535 The predecessor class exists when 
A does. (Contributed by Scott Fenton, 8-Feb-2011.)
 |-  A  e.  _V   =>    |- 
 Pred ( R ,  A ,  X )  e.  _V
 
Theoremcbvsetlike 23536* 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 23537* 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 23538 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 23539 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 23540 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 23541 Closed form of elpredim 23531. (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 23542 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 23543 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 23544 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 23545 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 23546 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 23547 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 23548 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 23549 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)
 |-  A. x  e.  A  Pred (  _E  ,  A ,  x )  e.  _V
 
Theoremsetlikess 23550* 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 23551* 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 23552 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 23553 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 23554 The predecessor class over 
(/) is always 
(/) (Contributed by Scott Fenton, 16-Apr-2011.)
 |-  Pred ( R ,  (/) ,  X )  =  (/)
 
Theorempreduz 23555 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  Pred (  <  ,  ( ZZ>= `  M ) ,  N )  =  ( M ... ( N  -  1 ) ) )
 
Theoremprednn 23556 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
 |-  ( N  e.  NN  ->  Pred
 (  <  ,  NN ,  N )  =  ( 1 ... ( N  -  1 ) ) )
 
Theoremprednn0 23557 The value of the predecessor class over  NN0. (Contributed by Scott Fenton, 9-May-2014.)
 |-  ( N  e.  NN0  ->  Pred (  <  ,  NN0 ,  N )  =  ( 0 ... ( N  -  1
 ) ) )
 
Theorempredfz 23558 Calculate the predecessor of an integer under a finite set of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  ( K  e.  ( M ... N )  ->  Pred (  <  ,  ( M ... N ) ,  K )  =  ( M ... ( K  -  1
 ) ) )
 
16.7.14  (Trans)finite Recursion Theorems
 
Theoremtfisg 23559* A closed form of tfis 4603. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( A. x  e.  On  ( A. y  e.  x  [. y  /  x ]. ph 
 ->  ph )  ->  A. x  e.  On  ph )
 
16.7.15  Well-founded induction
 
Theoremtz6.26 23560* All nonempty (possibly proper) subclasses of  A, which has a well-founded relation  R, have  R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  We  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  B  =/=  (/) ) ) 
 ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
 
Theoremtz6.26i 23561* All nonempty (possibly proper) subclasses of  A, which has a well-founded relation  R, have  R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   =>    |-  ( ( B 
 C_  A  /\  B  =/= 
 (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
 
Theoremwfi 23562* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if  B is a subclass of a well-ordered class  A with the property that every element of  B whose inital segment is included in 
A is itself equal to  A. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  We  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
 
Theoremwfii 23563* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if  B is a subclass of a well-ordered class  A with the property that every element of  B whose inital segment is included in 
A is itself equal to  A. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   =>    |-  ( ( B 
 C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  =  B )
 
Theoremwfisg 23564* Well-Founded Induction Schema. If a property passes from all elements less than  y of a well founded class  A to  y itself (induction hypothesis), then the property holds for all elements of  A. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( ( R  We  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremwfis 23565* Well-Founded Induction Schema. If all elements less than a given set  x of the well founded class  A have a property (induction hypothesis), then all elements of  A have that property. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremwfis2fg 23566* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   &    |-  ( y  e.  A  ->  ( A. z  e.  Pred  ( R ,  A ,  y
 ) ps  ->  ph )
 )   =>    |-  ( ( R  We  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremwfis2f 23567* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  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 )
 
Theoremwfis2g 23568* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( ( R  We  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremwfis2 23569* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremwfis3 23570* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  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 )
 
Theoremuzsinds 23571* Strong (or "total") induction principle over a set of upper integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  ( ZZ>= `  M )  ->  ( A. y  e.  ( M ... ( x  -  1
 ) ) ps  ->  ph ) )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ch )
 
Theoremnnsinds 23572* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  (
 1 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN  ->  ch )
 
Theoremnn0sinds 23573* Strong (or "total") induction principle over the non-negative integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN0  ->  ( A. y  e.  (
 0 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN0 
 ->  ch )
 
Theoremomsinds 23574* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  om  ->  ( A. y  e.  x  ps  ->  ph ) )   =>    |-  ( A  e.  om 
 ->  ch )
 
16.7.16  Transitive closure under a relationship
 
Syntaxctrpred 23575 Define the transitive predecessor class as a class.
 class  TrPred ( R ,  A ,  X )
 
Definitiondf-trpred 23576* Define the transitive predecessors of a class  X under a relationship  R and a class  A. This class can be thought of as the "smallest" class containing all elements of  A that are linked to  X by a chain of  R relationships (see trpredtr 23588 and trpredmintr 23589). Definition based off of Lemma 4.2 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory (check The Internet Archive for it now as Prof. Monk appears to have rewritten his website). (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  TrPred ( R ,  A ,  X )  =  U. ran  ( rec ( ( a  e. 
 _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om )
 
Theoremdftrpred2 23577* A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  TrPred ( R ,  A ,  X )  =  U_ i  e. 
 om  ( ( rec ( ( a  e. 
 _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om ) `  i )
 
Theoremtrpredeq1 23578 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  =  S  ->  TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
 
Theoremtrpredeq2 23579 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
 
Theoremtrpredeq3 23580 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
 
Theoremtrpredeq1d 23581 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( ph  ->  R  =  S )   =>    |-  ( ph  ->  TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
 
Theoremtrpredeq2d 23582 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
 
Theoremtrpredeq3d 23583 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( ph  ->  X  =  Y )   =>    |-  ( ph  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
 
Theoremeltrpred 23584* A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not really meant to be used directly: instead refer to trpredpred 23586 and trpredmintr 23589. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  ( Y  e.  TrPred ( R ,  A ,  X ) 
 <-> 
 E. i  e.  om  Y  e.  ( ( rec ( ( a  e. 
 _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om ) `  i ) )
 
Theoremtrpredlem1 23585* Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  ( Pred ( R ,  A ,  X )  e.  B  ->  ( ( rec (
 ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om ) `  i )  C_  A )
 
Theoremtrpredpred 23586 Assuming it exists, the predecessor class is a subset of the transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)
 |-  ( Pred ( R ,  A ,  X )  e.  B  -> 
 Pred ( R ,  A ,  X )  C_  TrPred ( R ,  A ,  X ) )
 
Theoremtrpredss 23587 The transitive predecessors form a subset of the base class. (Contributed by Scott Fenton, 20-Feb-2011.)
 |-  ( Pred ( R ,  A ,  X )  e.  B  -> 
 TrPred ( R ,  A ,  X )  C_  A )
 
Theoremtrpredtr 23588 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 23589* 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 23590 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 23591* 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 23592* 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 23593 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 23594 The class of transitive predecessors is empty when  A is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
 |-  TrPred ( R ,  (/) ,  X )  =  (/)
 
Theoremtrpredex 23595 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 23596* 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 ) ) )
 
16.7.17  Founded Induction
 
Theoremfrmin 23597* 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 23560 and tz7.5 4371. (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 23598* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 23597). 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 23562 and tfi 4602, 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 23599* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 23597). 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 23600* 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 )
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