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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembr1steq 24201 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 24202 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 24203 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 24204 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 )
 
18.7.10  Epsilon induction
 
Theoremsetinds 24205* 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 24206*  _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 24207*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
18.7.11  Ordinal numbers
 
Theoremelpotr 24208* 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 24209 Given ax-reg 7322, 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 24210 Lemma for dfon2 24219. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  Tr  U.
 { x  |  (
 ph  /\  Tr  x  /\  ps ) }
 
Theoremdfon2lem2 24211* Lemma for dfon2 24219 (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A
 
Theoremdfon2lem3 24212* Lemma for dfon2 24219. 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 24213* Lemma for dfon2 24219. 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 24214* Lemma for dfon2 24219. 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 24215* Lemma for dfon2 24219. 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 24216* Lemma for dfon2 24219. 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 24217* Lemma for dfon2 24219. 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 24218* Lemma for dfon2 24219. 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 24219*  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 24220 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
 |-  dom  _E  =  _V
 
Theoremrdgprc0 24221 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 24222 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 24223* 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 24224* Generalization of dfrdg2 24223 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 ) ) ) ) ) }
 
18.7.12  Defined equality axioms
 
Theoremaxextdfeq 24225 A version of ax-ext 2277 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 24226 A version of ax-13 1698 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 24227 ax-ext 2277 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 24228 axext4 2280 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 24229* 19.12vv 1851 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 24230 There is always a set not in  y. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  E. x  -.  x  e.  y
 
Theoremdistel 24231 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4208 and elirrv 7327.) (Contributed by Scott Fenton, 15-Dec-2010.)
 |-  ( -.  A. y  y  =  x  <->  -.  A. y  -.  x  e.  y )
 
Theoremaxextndbi 24232 axextnd 8229 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
 |-  E. z
 ( x  =  y  <-> 
 ( z  e.  x  <->  z  e.  y ) )
 
18.7.13  Hypothesis builders
 
Theoremhbntg 24233 A more general form of hbnt 1736. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. x ps )  ->  ( -.  ps  ->  A. x  -.  ph ) )
 
Theoremhbimtg 24234 A more general and closed form of hbim 1737. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
 ( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
 
Theoremhbaltg 24235 A more general and closed form of hbal 1722. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. y ps )  ->  ( A. x ph  ->  A. y A. x ps ) )
 
Theoremhbng 24236 A more general form of hbn 1732. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( -.  ps  ->  A. x  -.  ph )
 
Theoremhbimg 24237 A more general form of hbim 1737. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   &    |-  ( ch  ->  A. x th )   =>    |-  (
 ( ps  ->  ch )  ->  A. x ( ph  ->  th ) )
 
18.7.14  The Predecessor Class
 
Syntaxcpred 24238 The predecessors symbol.
 class  Pred ( R ,  A ,  X )
 
Definitiondf-pred 24239 Define the predecessor class of a relationship. This is the class of all elements  y of  A such that  y R X (see elpred 24248) . (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } )
 )
 
Theorempredeq1 24240 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  =  S  ->  Pred
 ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
 
Theorempredeq2 24241 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  Pred
 ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
 
Theorempredeq3 24242 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  Pred
 ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
 
Theorempredpredss 24243 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 24244 The predecessor class of  A is a subset of  A (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  Pred ( R ,  A ,  X )  C_  A
 
Theoremsspred 24245 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 24246* 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 24247 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 24248 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 24249 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 24250* 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 24251 The predecessor class exists when 
A does. (Contributed by Scott Fenton, 8-Feb-2011.)
 |-  A  e.  _V   =>    |- 
 Pred ( R ,  A ,  X )  e.  _V
 
Theoremcbvsetlike 24252* 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 24253* 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 24254 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 24255 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 24256 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 24257 Closed form of elpredim 24247. (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 24258 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 24259 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 24260 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 24261 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 24262 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 24263 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 24264 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 24265 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)
 |-  A. x  e.  A  Pred (  _E  ,  A ,  x )  e.  _V
 
Theoremsetlikess 24266* 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 24267* 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 24268 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 24269 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 24270 The predecessor class over 
(/) is always 
(/) (Contributed by Scott Fenton, 16-Apr-2011.)
 |-  Pred ( R ,  (/) ,  X )  =  (/)
 
Theorempreduz 24271 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 24272 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 24273 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 24274 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
 ) ) )
 
18.7.15  (Trans)finite Recursion Theorems
 
Theoremtfisg 24275* A closed form of tfis 4661. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( A. x  e.  On  ( A. y  e.  x  [. y  /  x ]. ph 
 ->  ph )  ->  A. x  e.  On  ph )
 
18.7.16  Well-founded induction
 
Theoremtz6.26 24276* 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 24277* 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 24278* 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 24279* 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 24280* 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 24281* 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 24282* 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 24283* 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 24284* 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 24285* 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 24286* 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 24287* 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 24288* 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 24289* 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 24290* 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 )
 
18.7.17  Transitive closure under a relationship
 
Syntaxctrpred 24291 Define the transitive predecessor class as a class.
 class  TrPred ( R ,  A ,  X )
 
Definitiondf-trpred 24292* 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 24304 and trpredmintr 24305). 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 24293* 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 24294 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  =  S  ->  TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
 
Theoremtrpredeq2 24295 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
 
Theoremtrpredeq3 24296 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
 
Theoremtrpredeq1d 24297 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 24298 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 24299 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 24300* 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 24302 and trpredmintr 24305. (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 ) )
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