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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembr2ndeq 25401 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremdfdm5 25402 Definition of domain in terms of and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrn5 25403 Definition of range in terms of and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsnnzb 25404 A singleton is non-empty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)

Theoremopelco3 25405 Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)

Theoremelima4 25406 Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)

19.7.16  Epsilon induction

Theoremsetinds 25407* 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 25408* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsetinds2 25409* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)

19.7.17  Ordinal numbers

Theoremelpotr 25410* A class of transitive sets is partially ordered by . (Contributed by Scott Fenton, 15-Oct-2010.)

Theoremdford5reg 25411 Given ax-reg 7562, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)

Theoremdfon2lem1 25412 Lemma for dfon2 25421. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem2 25413* Lemma for dfon2 25421 (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem3 25414* Lemma for dfon2 25421. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem4 25415* Lemma for dfon2 25421. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem5 25416* Lemma for dfon2 25421. Two sets satisfying the new definition also satisfy trichotomy with respect to (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem6 25417* Lemma for dfon2 25421. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem7 25418* Lemma for dfon2 25421. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem8 25419* Lemma for dfon2 25421. 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 25420* Lemma for dfon2 25421. A class of new ordinals is well-founded by . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremdfon2 25421* 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 25422 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)

Theoremrdgprc0 25423 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 25424 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 25425* 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 25426* Generalization of dfrdg2 25425 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

19.7.18  Defined equality axioms

Theoremaxextdfeq 25427 A version of ax-ext 2419 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremax13dfeq 25428 A version of ax-13 1728 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremaxextdist 25429 ax-ext 2419 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremaxext4dist 25430 axext4 2422 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theorem19.12b 25431* 19.12vv 1922 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremexnel 25432 There is always a set not in . (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremdistel 25433 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4383 and elirrv 7567.) (Contributed by Scott Fenton, 15-Dec-2010.)

Theoremaxextndbi 25434 axextnd 8468 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)

19.7.19  Hypothesis builders

Theoremhbntg 25435 A more general form of hbnt 1800. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimtg 25436 A more general and closed form of hbim 1837. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbaltg 25437 A more general and closed form of hbal 1752. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbng 25438 A more general form of hbn 1802. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimg 25439 A more general form of hbim 1837. (Contributed by Scott Fenton, 13-Dec-2010.)

19.7.20  The Predecessor Class

Syntaxcpred 25440 The predecessors symbol.

Definitiondf-pred 25441 Define the predecessor class of a relationship. This is the class of all elements of such that (see elpred 25454) . (Contributed by Scott Fenton, 29-Jan-2011.)

Theorempredeq123 25442 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)

Theorempredeq1 25443 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq2 25444 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq3 25445 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremnfpred 25446 Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)

Theorempredpredss 25447 If is a subset of , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredss 25448 The predecessor class of is a subset of (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremsspred 25449 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Theoremdfpred2 25450* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremdfpred3 25451* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremdfpred3g 25452* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremelpredim 25453 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)

Theoremelpred 25454 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)

Theoremelpredg 25455 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredreseq 25456* Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)

Theorempredasetex 25457 The predecessor class exists when does. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremcbvsetlike 25458* Change the bound variable in the statement stating that is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdffr4 25459* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredel 25460 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredpo 25461 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)

Theorempredso 25462 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredbrg 25463 Closed form of elpredim 25453. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

Theoremsetlikespec 25464 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 25465 Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredin 25466 Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredun 25467 Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempreddif 25468 Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)

Theorempredep 25469 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 25470 For an ordinal, the predecessor under and is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremepsetlike 25471 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremsetlikess 25472* If is set-like over , then it is set-like over any subclass of . (Contributed by Scott Fenton, 28-Mar-2011.)

Theorempreddowncl 25473* A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)

Theorempredpoirr 25474 Given a partial ordering, is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredfrirr 25475 Given a well-founded relationship, is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)

Theorempred0 25476 The predecessor class over is always (Contributed by Scott Fenton, 16-Apr-2011.)

Theorempreduz 25477 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)

Theoremprednn 25478 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)

Theoremprednn0 25479 The value of the predecessor class over . (Contributed by Scott Fenton, 9-May-2014.)

Theorempredfz 25480 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.)

19.7.21  (Trans)finite Recursion Theorems

Theoremtfisg 25481* A closed form of tfis 4836. (Contributed by Scott Fenton, 8-Jun-2011.)

19.7.22  Well-founded induction

Theoremtz6.26 25482* 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 25483* 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 25484* 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 25485* 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 25486* 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 25487* 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 25488* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis2f 25489* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis2g 25490* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis2 25491* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis3 25492* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremuzsinds 25493* Strong (or "total") induction principle over a set of upper integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnnsinds 25494* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnn0sinds 25495* Strong (or "total") induction principle over the non-negative integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremomsinds 25496* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)

19.7.23  Transitive closure under a relationship

Syntaxctrpred 25497 Define the transitive predecessor class as a class.

Definitiondf-trpred 25498* 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 25510 and trpredmintr 25511). 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 25499* A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremtrpredeq1 25500 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

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