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Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfrn5 23301 Definition of range in terms of and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

16.6.9  Epsilon induction

Theoremsetinds 23302* Principle of induction (set induction). If a property passes from all elements of to itself, then it holds for all . (Contributed by Scott Fenton, 10-Mar-2011.)

Theoremsetinds2f 23303* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsetinds2 23304* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)

16.6.10  Ordinal numbers

Theoremelpotr 23305* A class of transitive sets is partially ordered by . (Contributed by Scott Fenton, 15-Oct-2010.)

Theoremdford5reg 23306 Given ax-reg 7190, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)

Theoremdfon2lem1 23307 Lemma for dfon2 23316. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem2 23308* Lemma for dfon2 23316 (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem3 23309* Lemma for dfon2 23316. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem4 23310* Lemma for dfon2 23316. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem5 23311* Lemma for dfon2 23316. Two sets satisfying the new definition also satisfy trichotomy with respect to (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem6 23312* Lemma for dfon2 23316. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem7 23313* Lemma for dfon2 23316. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem8 23314* Lemma for dfon2 23316. The intersection of a non-empty class of new ordinals is itself a new ordinal and is contained within (Contributed by Scott Fenton, 26-Feb-2011.)

Theoremdfon2lem9 23315* Lemma for dfon2 23316. A class of new ordinals is well-founded by . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremdfon2 23316* 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.)

Theoremdomep 23317 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)

Theoremrdgprc0 23318 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.)

Theoremrdgprc 23319 The value of the recursive definition generator when is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrdg2 23320* Alternate definition of the recursive function generator when is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrdg3 23321* Generalization of dfrdg2 23320 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

16.6.11  Defined equality axioms

Theoremaxextdfeq 23322 A version of ax-ext 2234 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremax13dfeq 23323 A version of ax-13 1625 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremaxextdist 23324 ax-ext 2234 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremaxext4dist 23325 axext4 2237 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theorem19.12b 23326* 19.12vv 2031 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremexnel 23327 There is always a set not in . (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremdistel 23328 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4086 and elirrv 7195.) (Contributed by Scott Fenton, 15-Dec-2010.)

Theoremaxextndbi 23329 axextnd 8093 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)

16.6.12  Hypothesis builders

Theoremhbntg 23330 A more general form of hbnt 1717. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimtg 23331 A more general and closed form of hbim 1723. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbaltg 23332 A more general and closed form of hbal 1567. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbng 23333 A more general form of hbn 1722. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimg 23334 A more general form of hbim 1723. (Contributed by Scott Fenton, 13-Dec-2010.)

16.6.13  The Predecessor Class

Syntaxcpred 23335 The predecessors symbol.

Definitiondf-pred 23336 Define the predecessor class of a relationship. This is the class of all elements of such that (see elpred 23345) . (Contributed by Scott Fenton, 29-Jan-2011.)

Theorempredeq1 23337 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq2 23338 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq3 23339 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredpredss 23340 If is a subset of , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredss 23341 The predecessor class of is a subset of (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremsspred 23342 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Theoremdfpred2 23343* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremelpredim 23344 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)

Theoremelpred 23345 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)

Theoremelpredg 23346 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredreseq 23347* Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)

Theorempredasetex 23348 The predecessor class exists when does. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremcbvsetlike 23349* Change the bound variable in the statement stating that is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdffr4 23350* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredel 23351 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredpo 23352 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)

Theorempredso 23353 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredbrg 23354 Closed form of elpredim 23344. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

Theoremsetlikespec 23355 If is set-like in then all predecessors classes of elements of exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theorempredidm 23356 Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredin 23357 Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredun 23358 Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempreddif 23359 Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)

Theorempredep 23360 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.)

Theorempredon 23361 For an ordinal, the predecessor under and is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremepsetlike 23362 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremsetlikess 23363* If is set-like over , then it is set-like over any subclass of . (Contributed by Scott Fenton, 28-Mar-2011.)

Theorempreddowncl 23364* A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)

Theorempredpoirr 23365 Given a partial ordering, is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredfrirr 23366 Given a well-founded relationship, is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)

Theorempred0 23367 The predecessor class over is always (Contributed by Scott Fenton, 16-Apr-2011.)

Theorempreduz 23368 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)

Theoremprednn 23369 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)

Theoremprednn0 23370 The value of the predecessor class over . (Contributed by Scott Fenton, 9-May-2014.)

Theorempredfz 23371 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.)

16.6.14  (Trans)finite Recursion Theorems

Theoremtfisg 23372* A closed form of tfis 4536. (Contributed by Scott Fenton, 8-Jun-2011.)

16.6.15  Well-founded induction

Theoremtz6.26 23373* All nonempty (possibly proper) subclasses of , which has a well-founded relation , have -minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremtz6.26i 23374* All nonempty (possibly proper) subclasses of , which has a well-founded relation , have -minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfi 23375* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if is a subclass of a well-ordered class with the property that every element of whose inital segment is included in is itself equal to . (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfii 23376* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if is a subclass of a well-ordered class with the property that every element of whose inital segment is included in is itself equal to . (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfisg 23377* Well-Founded Induction Schema. If a property passes from all elements less than of a well founded class to itself (induction hypothesis), then the property holds for all elements of . (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis 23378* Well-Founded Induction Schema. If all elements less than a given set of the well founded class have a property (induction hypothesis), then all elements of have that property. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis2fg 23379* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis2f 23380* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis2g 23381* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis2 23382* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis3 23383* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremuzsinds 23384* Strong (or "total") induction principle over a set of upper integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnnsinds 23385* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnn0sinds 23386* Strong (or "total") induction principle over the non-negative integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremomsinds 23387* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)

16.6.16  Transitive closure under a relationship

Syntaxctrpred 23388 Define the transitive predecessor class as a class.

Definitiondf-trpred 23389* Define the transitive predecessors of a class under a relationship and a class . This class can be thought of as the "smallest" class containing all elements of that are linked to by a chain of relationships (see trpredtr 23401 and trpredmintr 23402). 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.)

Theoremdftrpred2 23390* A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremtrpredeq1 23391 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq2 23392 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq3 23393 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq1d 23394 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq2d 23395 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq3d 23396 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremeltrpred 23397* 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 23399 and trpredmintr 23402. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremtrpredlem1 23398* 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.)

Theoremtrpredpred 23399 Assuming it exists, the predecessor class is a subset of the transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)

Theoremtrpredss 23400 The transitive predecessors form a subset of the base class. (Contributed by Scott Fenton, 20-Feb-2011.)

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