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

Theoremgcd32 25401 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabsorb 25402 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

19.7.14  Properties of relationships

Theorembrtp 25403 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)

Theoremdftr6 25404 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)

Theoremcoep 25405* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremcoepr 25406* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremdffr5 25407 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremdfso2 25408 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)

Theoremdfpo2 25409 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)

Theorembr8 25410* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr6 25411* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr4 25412* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)

Theoremdfres3 25413 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcnvco1 25414 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremcnvco2 25415 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremeldm3 25416 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)

Theoremelrn3 25417 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)

Theorempocnv 25418 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremsocnv 25419 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)

19.7.15  Properties of functions and mappings

Theoremfunpsstri 25420 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)

Theoremfundmpss 25421 If a class is a proper subset of a function , then . (Contributed by Scott Fenton, 20-Apr-2011.)

Theoremfvresval 25422 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.)

Theoremmptrel 25423 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfunsseq 25424 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.)

Theoremfununiq 25425 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremfunbreq 25426 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremmpteq12d 25427 An equality inference for the maps to notation. Compare mpteq12dv 4312. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremfprb 25428* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)

Theorembr1steq 25429 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theorembr2ndeq 25430 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremdfdm5 25431 Definition of domain in terms of and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrn5 25432 Definition of range in terms of and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsnnzb 25433 A singleton is non-empty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)

Theoremopelco3 25434 Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)

Theoremelima4 25435 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 25436* 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 25437* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsetinds2 25438* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)

19.7.17  Ordinal numbers

Theoremelpotr 25439* A class of transitive sets is partially ordered by . (Contributed by Scott Fenton, 15-Oct-2010.)

Theoremdford5reg 25440 Given ax-reg 7589, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)

Theoremdfon2lem1 25441 Lemma for dfon2 25450. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem2 25442* Lemma for dfon2 25450 (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem3 25443* Lemma for dfon2 25450. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem4 25444* Lemma for dfon2 25450. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem5 25445* Lemma for dfon2 25450. Two sets satisfying the new definition also satisfy trichotomy with respect to (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem6 25446* Lemma for dfon2 25450. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem7 25447* Lemma for dfon2 25450. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem8 25448* Lemma for dfon2 25450. 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 25449* Lemma for dfon2 25450. A class of new ordinals is well-founded by . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremdfon2 25450* 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 25451 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)

Theoremrdgprc0 25452 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 25453 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 25454* 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 25455* Generalization of dfrdg2 25454 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 25456 A version of ax-ext 2423 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremax13dfeq 25457 A version of ax-13 1729 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremaxextdist 25458 ax-ext 2423 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremaxext4dist 25459 axext4 2426 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theorem19.12b 25460* 19.12vv 1924 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremexnel 25461 There is always a set not in . (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremdistel 25462 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4410 and elirrv 7594.) (Contributed by Scott Fenton, 15-Dec-2010.)

Theoremaxextndbi 25463 axextnd 8497 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)

19.7.19  Hypothesis builders

Theoremhbntg 25464 A more general form of hbnt 1801. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimtg 25465 A more general and closed form of hbim 1838. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbaltg 25466 A more general and closed form of hbal 1753. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbng 25467 A more general form of hbn 1803. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimg 25468 A more general form of hbim 1838. (Contributed by Scott Fenton, 13-Dec-2010.)

19.7.20  The Predecessor Class

Syntaxcpred 25469 The predecessors symbol.

Definitiondf-pred 25470 Define the predecessor class of a relationship. This is the class of all elements of such that (see elpred 25483) . (Contributed by Scott Fenton, 29-Jan-2011.)

Theorempredeq123 25471 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)

Theorempredeq1 25472 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq2 25473 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq3 25474 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremnfpred 25475 Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)

Theorempredpredss 25476 If is a subset of , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredss 25477 The predecessor class of is a subset of (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremsspred 25478 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Theoremdfpred2 25479* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremdfpred3 25480* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremdfpred3g 25481* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremelpredim 25482 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)

Theoremelpred 25483 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)

Theoremelpredg 25484 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredreseq 25485* Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)

Theorempredasetex 25486 The predecessor class exists when does. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremcbvsetlike 25487* Change the bound variable in the statement stating that is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdffr4 25488* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredel 25489 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredpo 25490 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)

Theorempredso 25491 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredbrg 25492 Closed form of elpredim 25482. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

Theoremsetlikespec 25493 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 25494 Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredin 25495 Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredun 25496 Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempreddif 25497 Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)

Theorempredep 25498 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 25499 For an ordinal, the predecessor under and is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremepsetlike 25500 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)

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