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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremr111 7401 The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
 |- 
 R1 : On -1-1-> _V
 
Theoremr1tr 7402 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.)
 |- 
 Tr  ( R1 `  A )
 
Theoremr1tr2 7403 The union of a cumulative hierarchy of sets at ordinal  A is a subset of the hierarchy at  A. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.)
 |- 
 U. ( R1 `  A )  C_  ( R1
 `  A )
 
Theoremr1ordg 7404 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.)
 |-  ( B  e.  dom  R1 
 ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
 
Theoremr1ord3g 7405 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)
 |-  ( ( A  e.  dom 
 R1  /\  B  e.  dom 
 R1 )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
 
Theoremr1ord 7406 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.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
 
Theoremr1ord2 7407 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 22-Sep-2003.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( R1 `  A )  C_  ( R1 `  B ) ) )
 
Theoremr1ord3 7408 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( R1 `  A )  C_  ( R1
 `  B ) ) )
 
Theoremr1sssuc 7409 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.)
 |-  ( A  e.  On  ->  ( R1 `  A )  C_  ( R1 `  suc  A ) )
 
Theoremr1pwss 7410 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  ( R1 `  B )  ->  ~P A  C_  ( R1
 `  B ) )
 
Theoremr1sscl 7411 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  ( R1 `  B ) 
 /\  C  C_  A )  ->  C  e.  ( R1 `  B ) )
 
Theoremr1val1 7412* 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.)
 |-  ( A  e.  dom  R1 
 ->  ( R1 `  A )  =  U_ x  e.  A  ~P ( R1
 `  x ) )
 
Theoremtz9.12lem1 7413* Lemma for tz9.12 7416. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   &    |-  F  =  ( z  e.  _V  |->  |^|
 { v  e.  On  |  z  e.  ( R1 `  v ) }
 )   =>    |-  ( F " A )  C_  On
 
Theoremtz9.12lem2 7414* Lemma for tz9.12 7416. (Contributed by NM, 22-Sep-2003.)
 |-  A  e.  _V   &    |-  F  =  ( z  e.  _V  |->  |^|
 { v  e.  On  |  z  e.  ( R1 `  v ) }
 )   =>    |- 
 suc  U. ( F " A )  e.  On
 
Theoremtz9.12lem3 7415* Lemma for tz9.12 7416. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   &    |-  F  =  ( z  e.  _V  |->  |^|
 { v  e.  On  |  z  e.  ( R1 `  v ) }
 )   =>    |-  ( A. x  e.  A  E. y  e. 
 On  x  e.  ( R1 `  y )  ->  A  e.  ( R1 ` 
 suc  suc  U. ( F " A ) ) )
 
Theoremtz9.12 7416* 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 7413 through tz9.12lem3 7415. (Contributed by NM, 22-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  E. y  e. 
 On  x  e.  ( R1 `  y )  ->  E. y  e.  On  A  e.  ( R1 `  y ) )
 
Theoremtz9.13 7417* 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.)
 |-  A  e.  _V   =>    |-  E. x  e. 
 On  A  e.  ( R1 `  x )
 
Theoremtz9.13g 7418* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 7417 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  e.  ( R1 `  x ) )
 
Theoremrankwflemb 7419* Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x ) )
 
Theoremrankf 7420 The domain and range of the  rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)
 |- 
 rank : U. ( R1 " On ) --> On
 
Theoremrankon 7421 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.)
 |-  ( rank `  A )  e.  On
 
Theoremr1elwf 7422 Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
 
Theoremrankvalb 7423* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7442 does not use Regularity, and so requires the assumption that  A is in the range of  R1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  A )  =  |^| { x  e. 
 On  |  A  e.  ( R1 `  suc  x ) } )
 
Theoremrankr1ai 7424 One direction of rankr1a 7462. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
 
Theoremrankvaln 7425 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 7439, unless  A is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 10-Sep-2013.)
 |-  ( -.  A  e.  U. ( R1 " On )  ->  ( rank `  A )  =  (/) )
 
Theoremrankidb 7426 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  A  e.  ( R1
 `  suc  ( rank `  A ) ) )
 
Theoremrankdmr1 7427 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( rank `  A )  e.  dom  R1
 
Theoremrankr1ag 7428 A version of rankr1a 7462 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B ) 
 <->  ( rank `  A )  e.  B ) )
 
Theoremrankr1bg 7429 A relationship between rank and  R1. See rankr1ag 7428 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B ) 
 <->  ( rank `  A )  C_  B ) )
 
Theoremr1rankidb 7430 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.)
 |-  ( A  e.  U. ( R1 " On )  ->  A  C_  ( R1 `  ( rank `  A )
 ) )
 
Theoremr1elssi 7431 The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 7432 that doesn't need  A to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  A  C_  U. ( R1 " On ) )
 
Theoremr1elss 7432 The range of the  R1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( A  e.  U. ( R1 " On ) 
 <->  A  C_  U. ( R1 " On ) )
 
Theorempwwf 7433 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.)
 |-  ( A  e.  U. ( R1 " On )  <->  ~P A  e.  U. ( R1 " On ) )
 
Theoremsswf 7434 A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  C_  A )  ->  B  e.  U. ( R1 " On )
 )
 
Theoremsnwf 7435 A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  { A }  e.  U. ( R1 " On ) )
 
Theoremunwf 7436 A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 ) 
 <->  ( A  u.  B )  e.  U. ( R1 " On ) )
 
Theoremprwf 7437 An unordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  { A ,  B }  e.  U. ( R1 " On )
 )
 
Theoremopwf 7438 An ordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  <. A ,  B >.  e.  U. ( R1 " On ) )
 
Theoremunir1 7439 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.)
 |- 
 U. ( R1 " On )  =  _V
 
Theoremjech9.3 7440 Every set belongs to some value of the cumulative hierarchy of sets function  R1, i.e. the indexed union of all values of 
R1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.)
 |-  U_ x  e.  On  ( R1 `  x )  =  _V
 
Theoremrankwflem 7441* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 7418 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  e.  ( R1 ` 
 suc  x ) )
 
Theoremrankval 7442* 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.)
 |-  A  e.  _V   =>    |-  ( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }
 
Theoremrankvalg 7443* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7442 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)
 |-  ( A  e.  V  ->  ( rank `  A )  =  |^| { x  e. 
 On  |  A  e.  ( R1 `  suc  x ) } )
 
Theoremrankval2 7444* Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.)
 |-  ( A  e.  B  ->  ( rank `  A )  =  |^| { x  e. 
 On  |  A  C_  ( R1 `  x ) } )
 
Theoremuniwf 7445 A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  <->  U. A  e.  U. ( R1 " On ) )
 
Theoremrankr1clem 7446 Lemma for rankr1c 7447. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( -.  A  e.  ( R1 `  B )  <->  B  C_  ( rank `  A ) ) )
 
Theoremrankr1c 7447 A relationship between the rank function and the cumulative hierarchy of sets function  R1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( B  =  (
 rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B ) ) ) )
 
Theoremrankidn 7448 A relationship between the rank function and the cumulative hierarchy of sets function  R1. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) )
 
Theoremrankpwi 7449 The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  ~P A )  =  suc  ( rank `  A ) )
 
Theoremrankelb 7450 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.)
 |-  ( B  e.  U. ( R1 " On )  ->  ( A  e.  B  ->  ( rank `  A )  e.  ( rank `  B )
 ) )
 
Theoremwfelirr 7451 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 7266. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  U. ( R1 " On )  ->  -.  A  e.  A )
 
Theoremrankval3b 7452* 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.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  A )  =  |^| { x  e. 
 On  |  A. y  e.  A  ( rank `  y
 )  e.  x }
 )
 
Theoremranksnb 7453 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  { A }
 )  =  suc  ( rank `  A ) )
 
Theoremrankonidlem 7454 Lemma for rankonid 7455. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
 |-  ( A  e.  dom  R1 
 ->  ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  A ) )
 
Theoremrankonid 7455 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.)
 |-  ( A  e.  dom  R1  <->  (
 rank `  A )  =  A )
 
Theoremonwf 7456 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 On  C_  U. ( R1 " On )
 
Theoremonssr1 7457 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.)
 |-  ( A  e.  dom  R1 
 ->  A  C_  ( R1 `  A ) )
 
Theoremrankr1g 7458 A relationship between the rank function and the cumulative hierarchy of sets function  R1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  V  ->  ( B  =  (
 rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B ) ) ) )
 
Theoremrankid 7459 Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   =>    |-  A  e.  ( R1 `  suc  ( rank `  A ) )
 
Theoremrankr1 7460 A relationship between the rank function and the cumulative hierarchy of sets function  R1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( B  =  ( rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 ` 
 suc  B ) ) )
 
Theoremssrankr1 7461 A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets  R1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( B  e.  On  ->  ( B  C_  ( rank `  A )  <->  -.  A  e.  ( R1
 `  B ) ) )
 
Theoremrankr1a 7462 A relationship between rank and  R1, clearly equivalent to ssrankr1 7461 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7490 for the subset verion. (Contributed by Raph Levien, 29-May-2004.)
 |-  A  e.  _V   =>    |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <-> 
 ( rank `  A )  e.  B ) )
 
Theoremr1val2 7463* 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.)
 |-  ( A  e.  On  ->  ( R1 `  A )  =  { x  |  ( rank `  x )  e.  A } )
 
Theoremr1val3 7464* 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.)
 |-  ( A  e.  On  ->  ( R1 `  A )  =  U_ x  e.  A  ~P { y  |  ( rank `  y )  e.  x } )
 
Theoremrankel 7465 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.)
 |-  B  e.  _V   =>    |-  ( A  e.  B  ->  ( rank `  A )  e.  ( rank `  B ) )
 
Theoremrankval3 7466* 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.)
 |-  A  e.  _V   =>    |-  ( rank `  A )  =  |^| { x  e.  On  |  A. y  e.  A  ( rank `  y
 )  e.  x }
 
Theorembndrank 7467* Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
 |-  ( E. x  e. 
 On  A. y  e.  A  ( rank `  y )  C_  x  ->  A  e.  _V )
 
Theoremunbndrank 7468* The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
 |-  ( -.  A  e.  _V 
 ->  A. x  e.  On  E. y  e.  A  x  e.  ( rank `  y )
 )
 
Theoremrankpw 7469 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.)
 |-  A  e.  _V   =>    |-  ( rank `  ~P A )  =  suc  ( rank `  A )
 
Theoremranklim 7470 The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)
 |-  ( Lim  B  ->  ( ( rank `  A )  e.  B  <->  ( rank `  ~P A )  e.  B ) )
 
Theoremr1pw 7471 A stronger property of  R1 than rankpw 7469. The latter merely proves that  R1 of the successor is a power set, but here we prove that if  A is in the cumulative hierarchy, then  ~P A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
 
Theoremr1pwOLD 7472 A stronger property of  R1 than rankpw 7469. The latter merely proves that  R1 of the successor is a power set, but here we prove that if  A is in the cumulative hierarchy, then  ~P A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( B  e.  On  ->  ( A  e.  ( R1 `  B )  <->  ~P A  e.  ( R1 `  suc  B ) ) )
 
Theoremr1pwcl 7473 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.)
 |-  ( Lim  B  ->  ( A  e.  ( R1
 `  B )  <->  ~P A  e.  ( R1 `  B ) ) )
 
Theoremrankssb 7474 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.)
 |-  ( B  e.  U. ( R1 " On )  ->  ( A  C_  B  ->  ( rank `  A )  C_  ( rank `  B )
 ) )
 
Theoremrankss 7475 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.)
 |-  B  e.  _V   =>    |-  ( A  C_  B  ->  ( rank `  A )  C_  ( rank `  B ) )
 
Theoremrankunb 7476 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.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  ( rank `  ( A  u.  B ) )  =  (
 ( rank `  A )  u.  ( rank `  B )
 ) )
 
Theoremrankprb 7477 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  ( rank ` 
 { A ,  B } )  =  suc  ( ( rank `  A )  u.  ( rank `  B ) ) )
 
Theoremrankopb 7478 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)
 |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On )
 )  ->  ( rank ` 
 <. A ,  B >. )  =  suc  suc  (
 ( rank `  A )  u.  ( rank `  B )
 ) )
 
Theoremrankuni2b 7479* 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.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  U. A )  =  U_ x  e.  A  ( rank `  x ) )
 
Theoremranksn 7480 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.)
 |-  A  e.  _V   =>    |-  ( rank `  { A } )  =  suc  ( rank `  A )
 
Theoremrankuni2 7481* 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.)
 |-  A  e.  _V   =>    |-  ( rank `  U. A )  =  U_ x  e.  A  ( rank `  x )
 
Theoremrankun 7482 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.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( rank `  ( A  u.  B ) )  =  ( ( rank `  A )  u.  ( rank `  B ) )
 
Theoremrankpr 7483 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.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( rank `  { A ,  B } )  =  suc  ( ( rank `  A )  u.  ( rank `  B ) )
 
Theoremrankop 7484 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.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( rank `  <. A ,  B >. )  =  suc  suc  ( ( rank `  A )  u.  ( rank `  B ) )
 
Theoremr1rankid 7485 Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  V  ->  A  C_  ( R1 `  ( rank `  A )
 ) )
 
Theoremrankeq0b 7486 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( A  =  (/)  <->  ( rank `  A )  =  (/) ) )
 
Theoremrankeq0 7487 A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( A  =  (/)  <->  (
 rank `  A )  =  (/) )
 
Theoremrankr1id 7488 The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  dom  R1  <->  (
 rank `  ( R1 `  A ) )  =  A )
 
Theoremrankuni 7489 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.)
 |-  ( rank `  U. A )  =  U. ( rank `  A )
 
Theoremrankr1b 7490 A relationship between rank and  R1. See rankr1a 7462 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( B  e.  On  ->  ( A  C_  ( R1 `  B )  <-> 
 ( rank `  A )  C_  B ) )
 
Theoremranksuc 7491 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( rank `  suc  A )  =  suc  ( rank `  A )
 
Theoremrankuniss 7492 Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.)
 |-  A  e.  _V   =>    |-  ( rank `  U. A )  C_  ( rank `  A )
 
Theoremrankval4 7493* 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.)
 |-  A  e.  _V   =>    |-  ( rank `  A )  =  U_ x  e.  A  suc  ( rank `  x )
 
Theoremrankbnd 7494* The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  suc  ( rank `  x )  C_  B  <->  (
 rank `  A )  C_  B )
 
Theoremrankbnd2 7495* The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( B  e.  On  ->  ( A. x  e.  A  ( rank `  x )  C_  B  <->  ( rank `  A )  C_  suc  B )
 )
 
Theoremrankc1 7496* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  ( rank `  x )  e.  ( rank ` 
 U. A )  <->  ( rank `  A )  =  ( rank ` 
 U. A ) )
 
Theoremrankc2 7497* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  A  ( rank `  x )  =  ( rank ` 
 U. A )  ->  ( rank `  A )  =  suc  ( rank `  U. A ) )
 
Theoremrankelun 7498 Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( ( rank `  A )  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D )
 )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D ) ) )
 
Theoremrankelpr 7499 Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( ( rank `  A )  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D )
 )  ->  ( rank ` 
 { A ,  B } )  e.  ( rank `  { C ,  D } ) )
 
Theoremrankelop 7500 Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( ( rank `  A )  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D )
 )  ->  ( rank ` 
 <. A ,  B >. )  e.  ( rank `  <. C ,  D >. ) )
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