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Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremen3lplem2 7301* Lemma for en3lp 7302. (Contributed by Alan Sare, 28-Oct-2011.)

Theoremen3lp 7302 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 27311 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremsetind 7303* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)

Theoremsetind2 7304 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)

Syntaxctc 7305 Extend class notation to include the transitive closure function.

Definitiondf-tc 7306* The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcvalg 7307* Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 7223; see tz9.1 7295.) (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcid 7308 Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtctr 7309 Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcmin 7310 Defining property of the transitive closure function: it is a subset of any transitive class containing . (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtc2 7311* A variant of the definition of the transitive closure function, using instead the smallest transitive set containing as a member, gives almost the same set, except that itself must be added because it is not usually a member of (and it is never a member if is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcsni 7312 The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)

Theoremtcss 7313 The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcel 7314 The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtcidm 7315 The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)

Theoremtc0 7316 The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)

Theoremtc00 7317 The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)

2.6.5  Rank

Syntaxcr1 7318 Extend class definition to include the cumulative hierarchy of sets function.

Syntaxcrnk 7319 Extend class definition to include rank function.

Definitiondf-r1 7320 Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 7347). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 7324, r1suc 7326, and r1lim 7328. Theorem r1val1 7342 shows a recursive definition that works for all values, and theorems r1val2 7393 and r1val3 7394 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), with a a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)

Definitiondf-rank 7321* Define the rank function. See rankval 7372, rankval2 7374, rankval3 7396, or rankval4 7423 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function : given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 7389 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 7392. (Contributed by NM, 11-Oct-2003.)

Theoremr1funlim 7322 The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 7323 avoids ax-rep 4028.) (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1fnon 7323 The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremr10 7324 Value of the cumulative hierarchy of sets function at . Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremr1sucg 7325 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1suc 7326 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremr1limg 7327* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1lim 7328* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremr1fin 7329 The first levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)

Theoremr1sdom 7330 Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)

Theoremr111 7331 The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)

Theoremr1tr 7332 The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremr1tr2 7333 The union of a cumulative hierarchy of sets at ordinal is a subset of the hierarchy at . JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.)

Theoremr1ordg 7334 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.)

Theoremr1ord3g 7335 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)

Theoremr1ord 7336 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremr1ord2 7337 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 22-Sep-2003.)

Theoremr1ord3 7338 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)

Theoremr1sssuc 7339 The value of the cumulative hierarchy of sets function is a subset of its value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.)

Theoremr1pwss 7340 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1sscl 7341 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1val1 7342* The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremtz9.12lem1 7343* Lemma for tz9.12 7346. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremtz9.12lem2 7344* Lemma for tz9.12 7346. (Contributed by NM, 22-Sep-2003.)

Theoremtz9.12lem3 7345* Lemma for tz9.12 7346. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremtz9.12 7346* A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 7343 through tz9.12lem3 7345. (Contributed by NM, 22-Sep-2003.)

Theoremtz9.13 7347* Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.)

Theoremtz9.13g 7348* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 7347 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)

Theoremrankwflemb 7349* Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremrankf 7350 The domain and range of the function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)

Theoremrankon 7351 The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 12-Sep-2013.)

Theoremr1elwf 7352 Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremrankvalb 7353* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7372 does not use Regularity, and so requires the assumption that is in the range of . (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremrankr1ai 7354 One direction of rankr1a 7392. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankvaln 7355 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 7369, unless is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremrankidb 7356 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Theoremrankdmr1 7357 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1ag 7358 A version of rankr1a 7392 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1bg 7359 A relationship between rank and . See rankr1ag 7358 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremr1rankidb 7360 Any set is a subset of the hierarchy of its rank. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremr1elssi 7361 The range of the function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7362 that doesn't need to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremr1elss 7362 The range of the function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theorempwwf 7363 A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremsswf 7364 A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremsnwf 7365 A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremunwf 7366 A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremprwf 7367 An unordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremopwf 7368 An ordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremunir1 7369 The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 8-Jun-2013.)

Theoremjech9.3 7370 Every set belongs to some value of the cumulative hierarchy of sets function , i.e. the indexed union of all values of is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.)

Theoremrankwflem 7371* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 7348 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)

Theoremrankval 7372* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremrankvalg 7373* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7372 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)

Theoremrankval2 7374* Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.)

Theoremuniwf 7375 A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1clem 7376 Lemma for rankr1c 7377. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1c 7377 A relationship between the rank function and the cumulative hierarchy of sets function . Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankidn 7378 A relationship between the rank function and the cumulative hierarchy of sets function . (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremrankpwi 7379 The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.)

Theoremrankelb 7380 The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremwfelirr 7381 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 7196. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremrankval3b 7382* The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremranksnb 7383 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremrankonidlem 7384 Lemma for rankonid 7385. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Theoremrankonid 7385 The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremonwf 7386 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremonssr1 7387 Initial segments of the ordinals are contained in initial segments of the cumulative hierarchy. (Contributed by FL, 20-Apr-2011.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1g 7388 A relationship between the rank function and the cumulative hierarchy of sets function . Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankid 7389 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1 7390 A relationship between the rank function and the cumulative hierarchy of sets function . Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

Theoremssrankr1 7391 A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets . Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1a 7392 A relationship between rank and , clearly equivalent to ssrankr1 7391 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7420 for the subset verion. (Contributed by Raph Levien, 29-May-2004.)

Theoremr1val2 7393* The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.)

Theoremr1val3 7394* The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankel 7395 The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankval3 7396* The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theorembndrank 7397* Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)

Theoremunbndrank 7398* The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)

Theoremrankpw 7399 The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremranklim 7400 The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)

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