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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrankonid 7501 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 7502 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 7503 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 7504 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 7505 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 7506 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 7507 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 7508 A relationship between rank and  R1, clearly equivalent to ssrankr1 7507 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 7536 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 7509* 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 7510* 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 7511 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 7512* 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 7513* 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 7514* 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 7515 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 7516 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 7517 A stronger property of  R1 than rankpw 7515. 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 7518 A stronger property of  R1 than rankpw 7515. 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 7519 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 7520 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 7521 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 7522 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 7523 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 7524 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 7525* 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 7526 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 7527* 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 7528 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 7529 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 7530 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 7531 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 7532 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 7533 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 7534 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 7535 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 7536 A relationship between rank and  R1. See rankr1a 7508 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 7537 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( rank `  suc  A )  =  suc  ( rank `  A )
 
Theoremrankuniss 7538 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 7539* 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 7540* 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 7541* 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 7542* 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 7543* 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 7544 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 7545 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 7546 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 7547 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 7548 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 7549 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 7552 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 7550 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 7551 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 7552 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 7549 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 7553 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 7554 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 7555* Scott's trick collects all sets that have a certain property and are of 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 7556* Scott's trick collects all sets that have a certain property and are of 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 7557* Theorem scheme version of scottex 7555. 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 7558* Theorem scheme version of scott0 7556. 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 7559* Lemma for the Collection Principle cp 7561. (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 7560* -Lemma for the Collection Principle cp 7561. (Contributed by NM, 17-Oct-2003.)
 |-  A  e.  _V   =>    |-  E. y A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  y )  =/=  (/) )
 
Theoremcp 7561* 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 7555 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 7562* A very strong generalization of the Axiom of Replacement (compare zfrep6 5748), derived from the Collection Principle cp 7561. 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 7563* A variant of the Boundedness Axiom bnd 7562 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 7564* The collection of all sets equinumerous to a set  A and having 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 7565* 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 least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 8173). 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 7564 justify the definition of kard. The restriction to 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 7566* 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 7567* 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 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 1666 and weth 8122, using scottexs 7557 to establish the existence of 
A.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 7566. (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 7568 Extend class definition to include the cardinal size function.
 class  card
 
Syntaxcale 7569 Extend class definition to include the aleph function.
 class  aleph
 
Syntaxccf 7570 Extend class definition to include the cofinality function.
 class  cf
 
Syntaxwacn 7571 The axiom of choice for limited-length sequences.
 class AC  A
 
Definitiondf-card 7572* 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 8168 for its value, cardval2 7624 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8173. 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 7573 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 7693, alephsuc 7695, and alephlim 7694. 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 7574* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 7873 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 7575* 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 7576* 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 7577 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 7578* 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 7579 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 7580 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  ~~  B  ->  ( A  e.  dom  card  <->  B  e.  dom  card ) )
 
Theoremfinnum 7581 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 7582 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  On  ->  A  e.  dom  card )
 
Theoremtskwe 7583* 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 7584 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
 )
 
Theoremcardval3 7585* An alternative definition of the value of  ( card `  A ) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  e.  dom  card 
 ->  ( card `  A )  =  |^| { x  e. 
 On  |  x  ~~  A } )
 
Theoremcardid2 7586 Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( A  e.  dom  card 
 ->  ( card `  A )  ~~  A )
 
Theoremisnum3 7587 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  dom  card  <->  (
 card `  A )  ~~  A )
 
Theoremoncardval 7588* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 8168, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( A  e.  On  ->  ( card `  A )  =  |^| { x  e. 
 On  |  x  ~~  A } )
 
Theoremoncardid 7589 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 8169, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
 
Theoremcardonle 7590 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  On  ->  ( card `  A )  C_  A )
 
Theoremcard0 7591 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
 |-  ( card `  (/) )  =  (/)
 
Theoremcardidm 7592 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( card `  ( card `  A ) )  =  ( card `  A )
 
Theoremoncard 7593* A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( E. x  A  =  ( card `  x )  <->  A  =  ( card `  A ) )
 
Theoremficardom 7594 The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)
 |-  ( A  e.  Fin  ->  ( card `  A )  e.  om )
 
Theoremficardid 7595 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
 |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
 
Theoremcardnn 7596 The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( A  e.  om  ->  ( card `  A )  =  A )
 
Theoremcardnueq0 7597 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
 |-  ( A  e.  dom  card 
 ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremcardne 7598 No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
 |-  ( A  e.  ( card `  B )  ->  -.  A  ~~  B )
 
Theoremcarden2a 7599 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 7600 are meant to replace carden 8173 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
 |-  ( ( ( card `  A )  =  (
 card `  B )  /\  ( card `  A )  =/= 
 (/) )  ->  A  ~~  B )
 
Theoremcarden2b 7600 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7599 are meant to replace carden 8173 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  ~~  B  ->  ( card `  A )  =  ( card `  B )
 )
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