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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem5recm6rec 24101 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
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18.7.6  Greatest common divisor and divisibility

Theorempdivsq 24102 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdspw 24103 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcd32 24104 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabsorb 24105 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

18.7.7  Properties of relationships

Theorembrtp 24106 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)

Theoremdftr6 24107 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)

Theoremcoep 24108* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremcoepr 24109* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremdffr5 24110 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremdfso2 24111 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)

Theoremdfpo2 24112 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)

Theorembr8 24113* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr6 24114* Substitution for an six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr4 24115* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)

Theoremdfres3 24116 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcnvco1 24117 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremcnvco2 24118 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremeldm3 24119 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)

Theoremelrn3 24120 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)

18.7.8  Properties of functions and mappings

Theoremfunpsstri 24121 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)

Theoremfundmpss 24122 If a class is a proper subset of a function , then . (Contributed by Scott Fenton, 20-Apr-2011.)

Theoremfvresval 24123 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 24124 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfunsseq 24125 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 24126 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremfunbreq 24127 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremmpteq12d 24128 An equality inference for the maps to notation. Compare mpteq12dv 4098. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremfprb 24129* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)

Theorembr1steq 24130 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theorembr2ndeq 24131 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremdfdm5 24132 Definition of domain in terms of and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrn5 24133 Definition of range in terms of and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

18.7.9  Epsilon induction

Theoremsetinds 24134* 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 24135* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsetinds2 24136* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)

18.7.10  Ordinal numbers

Theoremelpotr 24137* A class of transitive sets is partially ordered by . (Contributed by Scott Fenton, 15-Oct-2010.)

Theoremdford5reg 24138 Given ax-reg 7306, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)

Theoremdfon2lem1 24139 Lemma for dfon2 24148. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem2 24140* Lemma for dfon2 24148 (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem3 24141* Lemma for dfon2 24148. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem4 24142* Lemma for dfon2 24148. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem5 24143* Lemma for dfon2 24148. Two sets satisfying the new definition also satisfy trichotomy with respect to (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem6 24144* Lemma for dfon2 24148. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem7 24145* Lemma for dfon2 24148. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem8 24146* Lemma for dfon2 24148. 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 24147* Lemma for dfon2 24148. A class of new ordinals is well-founded by . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremdfon2 24148* 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 24149 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)

Theoremrdgprc0 24150 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 24151 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 24152* 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 24153* Generalization of dfrdg2 24152 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

18.7.11  Defined equality axioms

Theoremaxextdfeq 24154 A version of ax-ext 2264 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremax13dfeq 24155 A version of ax-13 1686 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremaxextdist 24156 ax-ext 2264 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremaxext4dist 24157 axext4 2267 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theorem19.12b 24158* 19.12vv 1839 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremexnel 24159 There is always a set not in . (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremdistel 24160 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4192 and elirrv 7311.) (Contributed by Scott Fenton, 15-Dec-2010.)

Theoremaxextndbi 24161 axextnd 8213 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)

18.7.12  Hypothesis builders

Theoremhbntg 24162 A more general form of hbnt 1724. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimtg 24163 A more general and closed form of hbim 1725. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbaltg 24164 A more general and closed form of hbal 1710. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbng 24165 A more general form of hbn 1720. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimg 24166 A more general form of hbim 1725. (Contributed by Scott Fenton, 13-Dec-2010.)

18.7.13  The Predecessor Class

Syntaxcpred 24167 The predecessors symbol.

Definitiondf-pred 24168 Define the predecessor class of a relationship. This is the class of all elements of such that (see elpred 24177) . (Contributed by Scott Fenton, 29-Jan-2011.)

Theorempredeq1 24169 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq2 24170 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq3 24171 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredpredss 24172 If is a subset of , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredss 24173 The predecessor class of is a subset of (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremsspred 24174 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Theoremdfpred2 24175* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremelpredim 24176 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)

Theoremelpred 24177 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)

Theoremelpredg 24178 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredreseq 24179* Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)

Theorempredasetex 24180 The predecessor class exists when does. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremcbvsetlike 24181* Change the bound variable in the statement stating that is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdffr4 24182* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredel 24183 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredpo 24184 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)

Theorempredso 24185 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredbrg 24186 Closed form of elpredim 24176. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

Theoremsetlikespec 24187 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.)
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Theorempredidm 24188 Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredin 24189 Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempredun 24190 Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)

Theorempreddif 24191 Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)

Theorempredep 24192 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 24193 For an ordinal, the predecessor under and is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremepsetlike 24194 The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)

Theoremsetlikess 24195* If is set-like over , then it is set-like over any subclass of . (Contributed by Scott Fenton, 28-Mar-2011.)

Theorempreddowncl 24196* A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)

Theorempredpoirr 24197 Given a partial ordering, is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredfrirr 24198 Given a well-founded relationship, is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)

Theorempred0 24199 The predecessor class over is always (Contributed by Scott Fenton, 16-Apr-2011.)

Theorempreduz 24200 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)

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