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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iunrelexp0 36201* | Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.) |
Theorem | relexpxpnnidm 36202 | Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.) |
Theorem | relexpiidm 36203 | Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.) |
Theorem | relexpss1d 36204 | The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.) |
Theorem | comptiunov2i 36205* | The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.) |
Theorem | corclrcl 36206 | The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.) |
Theorem | iunrelexpmin1 36207* | The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.) |
Theorem | relexpmulnn 36208 | With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.) |
Theorem | relexpmulg 36209 | With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.) |
Theorem | trclrelexplem 36210* | The union of relational powers to positive multiples of is a subset to the transitive closure raised to the power of . (Contributed by RP, 15-Jun-2020.) |
Theorem | iunrelexpmin2 36211* | The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.) |
Theorem | relexp01min 36212 | With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.) |
Theorem | relexp1idm 36213 | Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.) |
Theorem | relexp0idm 36214 | Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.) |
Theorem | relexp0a 36215 | Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.) |
Theorem | relexpxpmin 36216 | The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.) |
Theorem | relexpaddss 36217 | The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where is a relation as shown by relexpaddd 13054 or when the sum of the powers isn't 1 as shown by relexpaddg 13053. (Contributed by RP, 3-Jun-2020.) |
Theorem | iunrelexpuztr 36218* | The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 13060. (Contributed by RP, 4-Jun-2020.) |
Theorem | dftrcl3 36219* | Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.) |
Theorem | brfvtrcld 36220* | If two elements are connected by the transitive closure of a relation, then they are connected via instances the relation, for some counting number . (Contributed by RP, 22-Jul-2020.) |
Theorem | fvtrcllb1d 36221 | A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.) |
Theorem | trclfvcom 36222 | The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.) |
Theorem | cnvtrclfv 36223 | The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.) |
Theorem | cotrcltrcl 36224 | The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.) |
Theorem | trclimalb2 36225 | Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.) |
Theorem | brtrclfv2 36226* | Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.) |
Theorem | trclfvdecomr 36227 | The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.) |
Theorem | trclfvdecoml 36228 | The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.) |
Theorem | dmtrclfvRP 36229 | The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.) |
Theorem | rntrclfvRP 36230 | The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.) |
Theorem | rntrclfv 36231 | The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.) |
Theorem | dfrtrcl3 36232* | Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 13062. (Contributed by RP, 5-Jun-2020.) |
Theorem | brfvrtrcld 36233* | If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via instances the relation, for some natural number . Similar of dfrtrclrec2 13057. (Contributed by RP, 22-Jul-2020.) |
Theorem | fvrtrcllb0d 36234 | A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.) |
Theorem | fvrtrcllb0da 36235 | A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.) |
Theorem | fvrtrcllb1d 36236 | A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.) |
Theorem | dfrtrcl4 36237 | Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.) |
Theorem | corcltrcl 36238 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.) |
Theorem | cortrcltrcl 36239 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
Theorem | corclrtrcl 36240 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
Theorem | cotrclrcl 36241 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.) |
Theorem | cortrclrcl 36242 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
Theorem | cotrclrtrcl 36243 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
Theorem | cortrclrtrcl 36244 | The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.) |
Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879]. | ||
Theorem | frege77d 36245 | If the images of both and are subsets of and follows in the transitive closure of , then is an element of . Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 36443. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege81d 36246 | If the image of is a subset , is an element of and follows in the transitive closure of , then is an element of . Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 36447. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege83d 36247 | If the image of the union of and is a subset of the union of and , is an element of and follows in the transitive closure of , then is an element of the union of and . Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 36449. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege96d 36248 | If follows in the transitive closure of and follows in , then follows in the transitive closure of . Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 36462. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege87d 36249 | If the images of both and are subsets of and follows in the transitive closure of and follows in , then is an element of . Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 36453. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege91d 36250 | If follows in then follows in the transitive closure of . Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 36457. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege97d 36251 | If contains all elements after those in in the transitive closure of , then the image under of is a subclass of . Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 36463. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege98d 36252 | If follows and follows in the transitive closure of , then follows in the transitive closure of . Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 36464. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege102d 36253 | If either and are the same or follows in the transitive closure of and is the successor to , then follows in the transitive closure of . Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 36468. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege106d 36254 | If follows in , then either and are the same or follows in . Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 36472. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege108d 36255 | If either and are the same or follows in the transitive closure of and is the successor to , then either and are the same or follows in the transitive closure of . Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 36474. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege109d 36256 | If contains all elements of and all elements after those in in the transitive closure of , then the image under of is a subclass of . Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 36475. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege114d 36257 | If either relates and or and are the same, then either and are the same, relates and , relates and . Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 36480. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege111d 36258 | If either and are the same or follows in the transitive closure of and is the successor to , then either and are the same or follows or and in the transitive closure of . Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 36477. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege122d 36259 | If is a function, is the successor of , and is the successor of , then and are the same (or follows in the transitive closure of ). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 36488. (Contributed by RP, 15-Jul-2020.) |
Theorem | frege124d 36260 | If is a function, is the successor of , and follows in the transitive closure of , then and are the same or follows in the transitive closure of . Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 36490. (Contributed by RP, 16-Jul-2020.) |
Theorem | frege126d 36261 | If is a function, is the successor of , and follows in the transitive closure of , then (for distinct and ) either follows or follows in the transitive closure of . Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 36492. (Contributed by RP, 16-Jul-2020.) |
Theorem | frege129d 36262 | If is a function and (for distinct and ) either follows or follows in the transitive closure of , the successor of is either or it follows or it comes before in the transitive closure of . Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 36495. (Contributed by RP, 16-Jul-2020.) |
Theorem | frege131d 36263 | If is a function and contains all elements of and all elements before or after those elements of in the transitive closure of , then the image under of is a subclass of . Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 36497. (Contributed by RP, 17-Jul-2020.) |
Theorem | frege133d 36264 | If is a function and and both follow in the transitive closure of , then (for distinct and ) either follows or follows in the transitive closure of (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 36499. (Contributed by RP, 18-Jul-2020.) |
In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3234 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 36293, ax-frege2 36294, ax-frege8 36312, ax-frege28 36333, ax-frege31 36337, ax-frege41 36348, frege52 (see ax-frege52a 36360, frege52b 36392, and ax-frege52c 36391 for translations), frege54 (see ax-frege54a 36365, frege54b 36396 and ax-frege54c 36395 for translations) and frege58 (see ax-frege58a 36378, ax-frege58b 36404 and frege58c 36424 for translations) are considered "core" or axioms. However, at least ax-frege8 36312 can be derived from ax-frege1 36293 and ax-frege2 36294, see axfrege8 36310. Frege introduced implication, negation and the universal qualifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 36360, frege52b 36392, and ax-frege52c 36391. In dffrege69 36435, Frege introduced hereditary to say that relation , when restricted to operate on elements of class , will only have elements of class in its domain; see df-he 36275 for a definition in terms of image and subset. In dffrege76 36442, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write , which requires to also be a set. In dffrege99 36465, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write , which is a superclass of sets related by the reflexive-transitive relation . Finally, in dffrege115 36481, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write (to ignore any non-relational content of the class ). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html for details. English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 36245 for an example. | ||
Section 2 introduces the turnstile which turns an idea which may be true into an assertion that it does hold true . Section 5 introduces implication, . Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or , and , and two for exclusive-or corresponding to df-or 371, df-an 372, dfxor4 36265, dfxor5 36266. Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication or class equality in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f() is interpreted to mean if- where the content of the "function" is specified by the latter two argments or logical equivalent, while g() is read as and h( ) as . This necessarily introduces a need for set theory as both and cannot hold unless is a set. (Also .) Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f() as if- would result in the translation of f () as . For collections, we must generalize over set variables or run into the same problems; this leads to g() being translated as and so forth. Under this interpreation the text of section 11 gives us sp 1914 (or simpl 458 and simpr 462 and anifp 1428 in the propositional case) and statments similar to cbvalivw 1842, ax-gen 1663, alrimiv 1767, and alrimdv 1769. These last four introduce a generality and have no useful definition in terms of propositional variables. Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, , alex 1692, eqv 3714; Some are not B, , exnal 1693, pssv 3770, nev 36269; There are no C, , alnex 1659, eq0 3713; There exist D, , df-ex 1658, 0pss 3768, n0 3707. Notation for relations between expressions also can be written in various ways. All E are P, , dfss6 36270, df-ss 3386, dfss2 3389; No F are P, , alinexa 1707, disj1 3773; Some G are not P, , exanali 1715, nssinpss 3641, nss 3458; Some H are P, , bj-exnalimn 31157, 0pssin 36272, ndisj 36271. | ||
Theorem | dfxor4 36265 | Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
Theorem | dfxor5 36266 | Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
Theorem | df3or2 36267 | Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.) |
Theorem | df3an2 36268 | Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.) |
Theorem | nev 36269* | Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.) |
Theorem | dfss6 36270* | Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.) |
Theorem | ndisj 36271* | Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.) |
Theorem | 0pssin 36272* | Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.) |
The statement hereditary means relation is hereditary (in the sense of Frege) in the class or . The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked. As Frege wasn't using the language of classes or sets, this naturally differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on. | ||
Theorem | rp-imass 36273 | If the -image of a class is a subclass of , then the restriction of to is a subset of the Cartesian product of and . (Contributed by Richard Penner, 24-Dec-2019.) |
Syntax | whe 36274 | The property of relation being hereditary in class . |
hereditary | ||
Definition | df-he 36275 | The property of relation being hereditary in class . (Contributed by RP, 27-Mar-2020.) |
hereditary | ||
Theorem | dfhe2 36276 | The property of relation being hereditary in class . (Contributed by RP, 27-Mar-2020.) |
hereditary | ||
Theorem | dfhe3 36277* | The property of relation being hereditary in class . (Contributed by RP, 27-Mar-2020.) |
hereditary | ||
Theorem | heeq12 36278 | Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
hereditary hereditary | ||
Theorem | heeq1 36279 | Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
hereditary hereditary | ||
Theorem | heeq2 36280 | Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
hereditary hereditary | ||
Theorem | sbcheg 36281 | Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.) |
hereditary hereditary | ||
Theorem | hess 36282 | Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.) |
hereditary hereditary | ||
Theorem | xphe 36283 | Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.) |
hereditary | ||
Theorem | xpheOLD 36284 | Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) Obsolete version of xphe 36283 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
hereditary | ||
Theorem | 0he 36285 | The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) |
hereditary | ||
Theorem | 0heALT 36286 | The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
hereditary | ||
Theorem | he0 36287 | Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.) |
hereditary | ||
Theorem | unhe1 36288 | The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.) |
hereditary hereditary hereditary | ||
Theorem | snhesn 36289 | Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.) |
hereditary | ||
Theorem | idhe 36290 | The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.) |
hereditary | ||
Theorem | psshepw 36291 | The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
[] hereditary | ||
Theorem | sshepw 36292 | The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
[] hereditary | ||
Axiom | ax-frege1 36293 | The case in which is denied, is affirmed, and is affirmed is excluded. This is evident since cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
Axiom | ax-frege2 36294 | If a proposition is a necessary consequence of two propositions and and one of those, , is in turn a necessary consequence of the other, , then the proposition is a necessary consequence of the latter one, , alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
Theorem | rp-simp2-frege 36295 | Simplification of triple conjunction. Compare with simp2 1006. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Theorem | rp-simp2 36296 | Simplification of triple conjunction. Identical to simp2 1006. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Theorem | rp-frege3g 36297 |
Add antecedent to ax-frege2 36294. More general statement than frege3 36298.
Like ax-frege2 36294, it is essentially a closed form of mpd 15,
however it
has an extra antecedent.
It would be more natural to prove from a1i 11 and ax-frege2 36294 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Theorem | frege3 36298 | Add antecedent to ax-frege2 36294. Special case of rp-frege3g 36297. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Theorem | rp-misc1-frege 36299 | Double-use of ax-frege2 36294. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Theorem | rp-frege24 36300 | Introducing an embedded antecedent. Alternate proof for frege24 36318. Closed form for a1d 26. (Contributed by RP, 24-Dec-2019.) |
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