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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr111 7701 The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)

Theoremr1tr 7702 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 7703 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 7704 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 7705 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 7706 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 7707 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 7708 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 7709 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 7710 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1sscl 7711 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremr1val1 7712* 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 7713* Lemma for tz9.12 7716. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremtz9.12lem2 7714* Lemma for tz9.12 7716. (Contributed by NM, 22-Sep-2003.)

Theoremtz9.12lem3 7715* Lemma for tz9.12 7716. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremtz9.12 7716* 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 7713 through tz9.12lem3 7715. (Contributed by NM, 22-Sep-2003.)

Theoremtz9.13 7717* 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 7718* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 7717 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)

Theoremrankwflemb 7719* Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremrankf 7720 The domain and range of the function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)

Theoremrankon 7721 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 7722 Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremrankvalb 7723* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7742 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 7724 One direction of rankr1a 7762. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankvaln 7725 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 7739, unless is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 10-Sep-2013.)

Theoremrankidb 7726 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Theoremrankdmr1 7727 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1ag 7728 A version of rankr1a 7762 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1bg 7729 A relationship between rank and . See rankr1ag 7728 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremr1rankidb 7730 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 7731 The range of the function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7732 that doesn't need to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremr1elss 7732 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 7733 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 7734 A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremsnwf 7735 A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Theoremunwf 7736 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 7737 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 7738 An ordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremunir1 7739 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 7740 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 7741* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 7718 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)

Theoremrankval 7742* 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 7743* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7742 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)

Theoremrankval2 7744* Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.)

Theoremuniwf 7745 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 7746 Lemma for rankr1c 7747. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1c 7747 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 7748 A relationship between the rank function and the cumulative hierarchy of sets function . (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremrankpwi 7749 The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.)

Theoremrankelb 7750 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 7751 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 7566. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremrankval3b 7752* 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 7753 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremrankonidlem 7754 Lemma for rankonid 7755. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Theoremrankonid 7755 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 7756 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremonssr1 7757 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 7758 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 7759 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1 7760 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 7761 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 7762 A relationship between rank and , clearly equivalent to ssrankr1 7761 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7790 for the subset verion. (Contributed by Raph Levien, 29-May-2004.)

Theoremr1val2 7763* 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 7764* 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 7765 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 7766* 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 7767* Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)

Theoremunbndrank 7768* The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)

Theoremrankpw 7769 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 7770 The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)

Theoremr1pw 7771 A stronger property of than rankpw 7769. The latter merely proves that of the successor is a power set, but here we prove that if is in the cumulative hierarchy, then is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremr1pwOLD 7772 A stronger property of than rankpw 7769. The latter merely proves that of the successor is a power set, but here we prove that if is in the cumulative hierarchy, then is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremr1pwcl 7773 The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

Theoremrankssb 7774 The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankss 7775 The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankunb 7776 The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankprb 7777 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremrankopb 7778 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)

Theoremrankuni2b 7779* 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, 8-Jun-2013.)

Theoremranksn 7780 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankuni2 7781* The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankun 7782 The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankpr 7783 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankop 7784 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremr1rankid 7785 Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankeq0b 7786 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremrankeq0 7787 A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1id 7788 The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankuni 7789 The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankr1b 7790 A relationship between rank and . See rankr1a 7762 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremranksuc 7791 The rank of a successor. (Contributed by NM, 18-Sep-2006.)

Theoremrankuniss 7792 Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.)

Theoremrankval4 7793* The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.)

Theoremrankbnd 7794* The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.)

Theoremrankbnd2 7795* The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.)

Theoremrankc1 7796* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)

Theoremrankc2 7797* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)

Theoremrankelun 7798 Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

Theoremrankelpr 7799 Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremrankelop 7800 Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)

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