HomeHome Metamath Proof Explorer
Theorem List (p. 76 of 328)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunir1 7501 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 7502 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 7503* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 7480 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 7504* 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 7505* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7504 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 7506* 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 7507 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 7508 Lemma for rankr1c 7509. (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 7509 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 7510 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 7511 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 7512 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 7513 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 7328. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  e.  U. ( R1 " On )  ->  -.  A  e.  A )
 
Theoremrankval3b 7514* 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 7515 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 7516 Lemma for rankonid 7517. (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 7517 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 7518 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 7519 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 7520 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 7521 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 7522 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 7523 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 7524 A relationship between rank and  R1, clearly equivalent to ssrankr1 7523 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7552 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 7525* 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 7526* 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 7527 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 7528* 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 7529* 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 7530* 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 7531 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 7532 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 7533 A stronger property of  R1 than rankpw 7531. 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 7534 A stronger property of  R1 than rankpw 7531. 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 7535 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 7536 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 7537 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 7538 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 7539 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 7540 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 7541* 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 7542 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 7543* 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 7544 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 7545 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 7546 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 7547 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 7548 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 7549 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 7550 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 7551 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 7552 A relationship between rank and  R1. See rankr1a 7524 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 7553 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( rank `  suc  A )  =  suc  ( rank `  A )
 
Theoremrankuniss 7554 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 7555* 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 7556* 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 7557* 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 7558* 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 7559* 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 7560 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 7561 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 7562 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 >. ) )
 
Theoremrankxpl 7563 A lower bound on the rank of a cross product. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  X.  B )  =/=  (/)  ->  ( rank `  ( A  u.  B ) )  C_  ( rank `  ( A  X.  B ) ) )
 
Theoremrankxpu 7564 An upper bound on the rank of a cross product. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( rank `  ( A  X.  B ) )  C_  suc 
 suc  ( rank `  ( A  u.  B ) )
 
Theoremrankxplim 7565 The rank of a cross product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 7568 for the successor case. (Contributed by NM, 19-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( Lim  ( rank `  ( A  u.  B ) )  /\  ( A  X.  B )  =/=  (/) )  ->  ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B ) ) )
 
Theoremrankxplim2 7566 If the rank of a cross product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B ) ) )
 
Theoremrankxplim3 7567 The rank of a cross product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( Lim  ( rank `  ( A  X.  B ) )  <->  Lim  U. ( rank `  ( A  X.  B ) ) )
 
Theoremrankxpsuc 7568 The rank of a cross product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 7565 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( ( rank `  ( A  u.  B ) )  =  suc  C 
 /\  ( A  X.  B )  =/=  (/) )  ->  ( rank `  ( A  X.  B ) )  = 
 suc  suc  ( rank `  ( A  u.  B ) ) )
 
Theoremtcwf 7569 The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( TC `  A )  e.  U. ( R1 " On ) )
 
Theoremtcrank 7570 This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below  A. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below  ( rank `  A ), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of  ( TC `  A ) has a rank below the rank of  A, since intuitively it contains only the members of  A and the members of those and so on, but nothing "bigger" than  A. (Contributed by Mario Carneiro, 23-Jun-2013.)
 |-  ( A  e.  U. ( R1 " On )  ->  ( rank `  A )  =  ( rank " ( TC `  A ) ) )
 
2.6.6  Scott's trick; collection principle; Hilbert's epsilon
 
Theoremscottex 7571* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)
 |- 
 { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
 
Theoremscott0 7572* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e.  A is empty). (Contributed by NM, 15-Oct-2003.)
 |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  (/) )
 
Theoremscottexs 7573* Theorem scheme version of scottex 7571. The collection of all  x of minimum rank such that 
ph ( x ) is true, is a set. (Contributed by NM, 13-Oct-2003.)
 |- 
 { x  |  (
 ph  /\  A. y (
 [. y  /  x ].
 ph  ->  ( rank `  x )  C_  ( rank `  y
 ) ) ) }  e.  _V
 
Theoremscott0s 7574* Theorem scheme version of scott0 7572. The collection of all  x of minimum rank such that 
ph ( x ) is true, is not empty iff there is an  x such that  ph ( x ) holds. (Contributed by NM, 13-Oct-2003.)
 |-  ( E. x ph  <->  { x  |  ( ph  /\ 
 A. y ( [. y  /  x ]. ph  ->  (
 rank `  x )  C_  ( rank `  y )
 ) ) }  =/=  (/) )
 
Theoremcplem1 7575* Lemma for the Collection Principle cp 7577. (Contributed by NM, 17-Oct-2003.)
 |-  C  =  { y  e.  B  |  A. z  e.  B  ( rank `  y
 )  C_  ( rank `  z ) }   &    |-  D  =  U_ x  e.  A  C   =>    |- 
 A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  D )  =/=  (/) )
 
Theoremcplem2 7576* -Lemma for the Collection Principle cp 7577. (Contributed by NM, 17-Oct-2003.)
 |-  A  e.  _V   =>    |-  E. y A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )
 
Theoremcp 7577* Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 7571 that collapses a proper class into a set of minimum rank. The wff  ph can be thought of as  ph ( x ,  y ). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
 |- 
 E. w A. x  e.  z  ( E. y ph  ->  E. y  e.  w  ph )
 
Theorembnd 7578* A very strong generalization of the Axiom of Replacement (compare zfrep6 5764), derived from the Collection Principle cp 7577. Its strength lies in the rather profound fact that  ph ( x ,  y ) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.)
 |-  ( A. x  e.  z  E. y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
 
Theorembnd2 7579* A variant of the Boundedness Axiom bnd 7578 that picks a subset  z out of a possibly proper class 
B in which a property is true. (Contributed by NM, 4-Feb-2004.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. z
 ( z  C_  B  /\  A. x  e.  A  E. y  e.  z  ph ) )
 
Theoremkardex 7580* The collection of all sets equinumerous to a set  A and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)
 |- 
 { x  |  ( x  ~~  A  /\  A. y ( y  ~~  A  ->  ( rank `  x )  C_  ( rank `  y
 ) ) ) }  e.  _V
 
Theoremkarden 7581* If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 8189). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 7580 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from  { x  |  x  ~~  A }. (Contributed by NM, 18-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  =  { x  |  ( x  ~~  A  /\  A. y
 ( y  ~~  A  ->  ( rank `  x )  C_  ( rank `  y )
 ) ) }   &    |-  D  =  { x  |  ( x  ~~  B  /\  A. y ( y  ~~  B  ->  ( rank `  x )  C_  ( rank `  y
 ) ) ) }   =>    |-  ( C  =  D  <->  A  ~~  B )
 
Theoremhtalem 7582* Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional  R  We  A antecedent. The element  B is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  A  e.  _V   &    |-  B  =  ( iota_ x  e.  A A. y  e.  A  -.  y R x )   =>    |-  ( ( R  We  A  /\  A  =/=  (/) )  ->  B  e.  A )
 
Theoremhta 7583* A ZFC emulation of Hilbert's transfinite axiom. The set  B has the properties of Hilbert's epsilon, except that it also depends on a well-ordering  R. This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See http://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and http://us.metamath.org/downloads/megillaward2005he.pdf.

Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires  R  We  A as an antecedent. Class  A collects the sets of the least rank for which  ph ( x ) is true. Class  B, which emulates the epsilon, is the minimum element in a well-ordering  R on  A.

If a well-ordering  R on  A can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace  R with a dummy set variable, say  w, and attach  w  We  A as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point,  B (which will have  w as a free variable) will no longer be present, and we can eliminate  w  We  A by applying exlimiv 1624 and weth 8138, using scottexs 7573 to establish the existence of 
A.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 7582. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

 |-  A  =  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
 rank `  x )  C_  ( rank `  y )
 ) ) }   &    |-  B  =  ( iota_ z  e.  A A. w  e.  A  -.  w R z )   =>    |-  ( R  We  A  ->  ( ph  ->  [. B  /  x ]. ph )
 )
 
2.6.7  Cardinal numbers
 
Syntaxccrd 7584 Extend class definition to include the cardinal size function.
 class  card
 
Syntaxcale 7585 Extend class definition to include the aleph function.
 class  aleph
 
Syntaxccf 7586 Extend class definition to include the cofinality function.
 class  cf
 
Syntaxwacn 7587 The axiom of choice for limited-length sequences.
 class AC  A
 
Definitiondf-card 7588* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 8184 for its value, cardval2 7640 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8189. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
 |- 
 card  =  ( x  e.  _V  |->  |^| { y  e. 
 On  |  y  ~~  x } )
 
Definitiondf-aleph 7589 Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 7709, alephsuc 7711, and alephlim 7710. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.)
 |-  aleph  =  rec (har ,  om )
 
Definitiondf-cf 7590* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 7889 for its value and a description. (Contributed by NM, 1-Apr-2004.)
 |- 
 cf  =  ( x  e.  On  |->  |^| { y  |  E. z ( y  =  ( card `  z
 )  /\  ( z  C_  x  /\  A. v  e.  x  E. u  e.  z  v  C_  u ) ) } )
 
Definitiondf-acn 7591* Define a local and length-limited version of the axiom of choice. The definition of the predicate 
X  e. AC  A is that for all families of nonempty subsets of  X indexed on  A (i.e. functions  A --> ~P X  \  { (/) }), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |- AC  A  =  { x  |  ( A  e.  _V  /\ 
 A. f  e.  (
 ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
 )  e.  ( f `
  y ) ) }
 
Theoremcardf2 7592* The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |- 
 card : { x  |  E. y  e.  On  y  ~~  x } --> On
 
Theoremcardon 7593 The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( card `  A )  e.  On
 
Theoremisnum2 7594* A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A )
 
Theoremisnumi 7595 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card
 )
 
Theoremennum 7596 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  ~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
 
Theoremfinnum 7597 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  Fin  ->  A  e.  dom  card )
 
Theoremonenon 7598 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  On  ->  A  e.  dom  card )
 
Theoremtskwe 7599* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  { x  e. 
 ~P A  |  x  ~<  A }  C_  A )  ->  A  e.  dom  card
 )
 
Theoremxpnum 7600 The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  X.  B )  e.  dom  card
 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >