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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalephnbtwn2 8501 No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |- 
 -.  ( ( aleph `  A )  ~<  B  /\  B  ~<  ( aleph `  suc  A ) )
 
Theoremalephordilem1 8502 Lemma for alephordi 8503. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  On  ->  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
 
Theoremalephordi 8503 Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B )
 ) )
 
Theoremalephord 8504 Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B 
 <->  ( aleph `  A )  ~<  ( aleph `  B )
 ) )
 
Theoremalephord2 8505 Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B 
 <->  ( aleph `  A )  e.  ( aleph `  B )
 ) )
 
Theoremalephord2i 8506 Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( aleph `  A )  e.  ( aleph `  B )
 ) )
 
Theoremalephord3 8507 Ordering property of the aleph function. (Contributed by NM, 11-Nov-2003.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  ( aleph `  A )  C_  ( aleph `  B )
 ) )
 
Theoremalephsucdom 8508 A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B )  <->  A  ~<  ( aleph ` 
 suc  B ) ) )
 
Theoremalephsuc2 8509* An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 8059 function by transfinite recursion, starting from 
om. Using this theorem we could define the aleph function with  { z  e.  On  |  z  ~<_  x } in place of  |^| { z  e.  On  |  x 
~<  z } in df-aleph 8373. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e. 
 On  |  x  ~<_  (
 aleph `  A ) }
 )
 
Theoremalephdom 8510 Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  ( aleph `  A )  ~<_  ( aleph `  B )
 ) )
 
Theoremalephgeom 8511 Every aleph is greater than or equal to the set of natural numbers. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  On  <->  om  C_  ( aleph `  A )
 )
 
Theoremalephislim 8512 Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  On  <->  Lim  ( aleph `  A )
 )
 
Theoremaleph11 8513 The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A )  =  (
 aleph `  B )  <->  A  =  B ) )
 
Theoremalephf1 8514 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 8532. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  aleph : On -1-1-> On
 
Theoremalephsdom 8515 If an ordinal is smaller than an initial ordinal, it is strictly dominated by it. (Contributed by Jeff Hankins, 24-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  ( aleph `  B )  <->  A 
 ~<  ( aleph `  B )
 ) )
 
Theoremalephdom2 8516 A dominated initial ordinal is included. (Contributed by Jeff Hankins, 24-Oct-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A )  C_  B  <->  (
 aleph `  A )  ~<_  B ) )
 
Theoremalephle 8517 The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 8538, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.)
 |-  ( A  e.  On  ->  A  C_  ( aleph `  A ) )
 
Theoremcardaleph 8518* Given any transfinite cardinal number  A, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
 
Theoremcardalephex 8519* Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)
 |-  ( om  C_  A  ->  ( ( card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
 
Theoreminfenaleph 8520* An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A ) 
 ->  E. x  e.  ran  aleph x  ~~  A )
 
Theoremisinfcard 8521 Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph
 )
 
Theoremiscard3 8522 Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( card `  A )  =  A  <->  A  e.  ( om  u.  ran  aleph ) )
 
Theoremcardnum 8523 Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in  om plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.)
 |- 
 { x  |  (
 card `  x )  =  x }  =  ( om  u.  ran  aleph )
 
Theoremalephinit 8524* An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  A. x  e.  On  ( A  ~<_  x  ->  A 
 C_  x ) ) )
 
Theoremcarduniima 8525 The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
 |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A )  e.  ( om  u.  ran  aleph ) ) )
 
Theoremcardinfima 8526* If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
 |-  ( A  e.  B  ->  ( ( F : A
 --> ( om  u.  ran  aleph
 )  /\  E. x  e.  A  ( F `  x )  e.  ran  aleph
 )  ->  U. ( F
 " A )  e. 
 ran  aleph ) )
 
Theoremalephiso 8527 Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)
 |-  aleph 
 Isom  _E  ,  _E  ( On ,  { x  |  ( om  C_  x  /\  ( card `  x )  =  x ) } )
 
Theoremalephprc 8528 The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.)
 |- 
 -.  ran  aleph  e.  _V
 
Theoremalephsson 8529 The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
 |- 
 ran  aleph  C_  On
 
Theoremunialeph 8530 The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
 |- 
 U. ran  aleph  =  On
 
Theoremalephsmo 8531 The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
 |- 
 Smo  aleph
 
Theoremalephf1ALT 8532 Alternate proof of alephf1 8514. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  aleph : On -1-1-> On
 
Theoremalephfplem1 8533 Lemma for alephfp 8537. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( H `  (/) )  e. 
 ran  aleph
 
Theoremalephfplem2 8534* Lemma for alephfp 8537. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w ) ) )
 
Theoremalephfplem3 8535* Lemma for alephfp 8537. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( v  e.  om  ->  ( H `  v
 )  e.  ran  aleph )
 
Theoremalephfplem4 8536 Lemma for alephfp 8537. (Contributed by NM, 5-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |- 
 U. ( H " om )  e.  ran  aleph
 
Theoremalephfp 8537 The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 8538 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( aleph `  U. ( H
 " om ) )  =  U. ( H
 " om )
 
Theoremalephfp2 8538 The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 8537 for an actual example of a fixed point. Compare the inequality alephle 8517 that holds in general. Note that if  x is a fixed point, then  aleph `  aleph `  aleph ` ...  aleph `  x  =  x. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)
 |- 
 E. x  e.  On  ( aleph `  x )  =  x
 
Theoremalephval3 8539* An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)
 |-  ( A  e.  On  ->  ( aleph `  A )  =  |^| { x  |  ( ( card `  x )  =  x  /\  om  C_  x  /\  A. y  e.  A  -.  x  =  ( aleph `  y )
 ) } )
 
Theoremalephsucpw2 8540 The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9100 or gchaleph2 9096.) The transposed form alephsucpw 8993 cannot be proven without the AC, and is in fact equivalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |- 
 -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
 
Theoremmappwen 8541 Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A 
 ^m  B )  ~~  ~P B )
 
Theoremfinnisoeu 8542* A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E! f  f 
 Isom  _E  ,  R  ( ( card `  A ) ,  A ) )
 
Theoremiunfictbso 8543 Countability of a countable union of finite sets with a strict (not globally well) order fulfilling the choice role. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( ( A  ~<_  om  /\  A  C_  Fin  /\  B  Or  U. A )  ->  U. A  ~<_  om )
 
2.6.8  Axiom of Choice equivalents
 
Syntaxwac 8544 Wff for an abbreviation of the axiom of choice.
 wff CHOICE
 
Definitiondf-ac 8545* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 8887 as our definition, because the equivalence to more standard forms (dfac2 8559) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 8887 itself as dfac0 8561. (Contributed by Mario Carneiro, 22-Feb-2015.)

 |-  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
 )
 
Theoremaceq1 8546* Equivalence of two versions of the Axiom of Choice ax-ac 8887. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  <->  E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. x A. z ( E. x ( ( z  e.  w  /\  w  e.  x )  /\  (
 z  e.  x  /\  x  e.  y )
 ) 
 <->  z  =  x ) ) )
 
Theoremaceq0 8547* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 8887. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  <->  E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t
 ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y )
 ) 
 <->  u  =  v ) ) )
 
Theoremaceq2 8548* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  <->  E. y A. z  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  ( z  e.  v  /\  w  e.  v ) ) )
 
Theoremaceq3lem 8549* Lemma for dfac3 8550. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  F  =  ( w  e.  dom  y  |->  ( f `  { u  |  w y u }
 ) )   =>    |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  (
 f `  z )  e.  z )  ->  E. f
 ( f  C_  y  /\  f  Fn  dom  y
 ) )
 
Theoremdfac3 8550* Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
 |-  (CHOICE  <->  A. x E. f A. z  e.  x  (
 z  =/=  (/)  ->  (
 f `  z )  e.  z ) )
 
Theoremdfac4 8551* Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (CHOICE  <->  A. x E. f ( f  Fn  x  /\  A. z  e.  x  ( z  =/=  (/)  ->  (
 f `  z )  e.  z ) ) )
 
Theoremdfac5lem1 8552* Lemma for dfac5 8557. (Contributed by NM, 12-Apr-2004.)
 |-  ( E! v  v  e.  ( ( { w }  X.  w )  i^i  y )  <->  E! g ( g  e.  w  /\  <. w ,  g >.  e.  y
 ) )
 
Theoremdfac5lem2 8553* Lemma for dfac5 8557. (Contributed by NM, 12-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   =>    |-  ( <. w ,  g >.  e. 
 U. A  <->  ( w  e.  h  /\  g  e.  w ) )
 
Theoremdfac5lem3 8554* Lemma for dfac5 8557. (Contributed by NM, 12-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   =>    |-  (
 ( { w }  X.  w )  e.  A  <->  ( w  =/=  (/)  /\  w  e.  h ) )
 
Theoremdfac5lem4 8555* Lemma for dfac5 8557. (Contributed by NM, 11-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   &    |-  B  =  ( U. A  i^i  y )   &    |-  ( ph  <->  A. x ( (
 A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/= 
 w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) ) )   =>    |-  ( ph  ->  E. y A. z  e.  A  E! v  v  e.  ( z  i^i  y ) )
 
Theoremdfac5lem5 8556* Lemma for dfac5 8557. (Contributed by NM, 12-Apr-2004.)
 |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }   &    |-  B  =  ( U. A  i^i  y )   &    |-  ( ph  <->  A. x ( (
 A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/= 
 w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) ) )   =>    |-  ( ph  ->  E. f A. w  e.  h  ( w  =/=  (/)  ->  (
 f `  w )  e.  w ) )
 
Theoremdfac5 8557* Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x ( ( A. z  e.  x  z  =/= 
 (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/= 
 w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) ) )
 
Theoremdfac2a 8558* Our Axiom of Choice (in the form of ac3 8890) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8559 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( A. x E. y A. z  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
 z  e.  v  /\  w  e.  v )
 )  -> CHOICE )
 
Theoremdfac2 8559* Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 8890). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 8112 and preleq 8122 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 8558.) TODO: Fix label in comment, and put label changes into list at top of set.mm. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (CHOICE  <->  A. x E. y A. z  e.  x  (
 z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
 z  e.  v  /\  w  e.  v )
 ) )
 
Theoremdfac7 8560* Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 8889). The proof does not depend AC on but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )
 )
 
Theoremdfac0 8561* Equivalence of two versions of the Axiom of Choice. The proof uses the Axiom of Regularity. The right-hand side is our original ax-ac 8887. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y
 ) )  <->  u  =  v
 ) ) )
 
Theoremdfac1 8562* Equivalence of two versions of the Axiom of Choice ax-ac 8887. The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. x A. z
 ( E. x ( ( z  e.  w  /\  w  e.  x )  /\  ( z  e.  x  /\  x  e.  y ) )  <->  z  =  x ) ) )
 
Theoremdfac8 8563* A proof of the equivalency of the Well Ordering Theorem weth 8923 and the Axiom of Choice ac7 8901. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  (CHOICE  <->  A. x E. r  r  We  x )
 
Theoremdfac9 8564* Equivalence of the axiom of choice with a statement related to ac9 8911; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (CHOICE  <->  A. f ( ( Fun  f  /\  (/)  e/  ran  f )  ->  X_ x  e.  dom  f ( f `
  x )  =/=  (/) ) )
 
Theoremdfac10 8565 Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  (CHOICE  <->  dom  card  =  _V )
 
Theoremdfac10c 8566* Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  (CHOICE  <->  A. x E. y  e. 
 On  y  ~~  x )
 
Theoremdfac10b 8567 Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 8545). (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  (CHOICE  <->  ( 
 ~~  " On )  =  _V )
 
Theoremacacni 8568 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( (CHOICE 
 /\  A  e.  V )  -> AC  A  =  _V )
 
Theoremdfacacn 8569 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  (CHOICE  <->  A. xAC  x  =  _V )
 
Theoremdfac13 8570 The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  (CHOICE  <->  A. x  x  e. AC  x )
 
Theoremdfac12lem1 8571* Lemma for dfac12 8577. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  F : ~P (har `  ( R1 `  A ) ) -1-1-> On )   &    |-  G  = recs (
 ( x  e.  _V  |->  ( y  e.  ( R1 `  dom  x ) 
 |->  if ( dom  x  =  U. dom  x ,  ( ( suc  U. ran  U.
 ran  x  .o  ( rank `  y ) )  +o  ( ( x `
  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  (
 ( `'OrdIso (  _E  ,  ran  ( x `  U. dom  x ) )  o.  ( x `  U. dom  x ) ) " y
 ) ) ) ) ) )   &    |-  ( ph  ->  C  e.  On )   &    |-  H  =  ( `'OrdIso (  _E  ,  ran  ( G `  U. C ) )  o.  ( G `  U. C ) )   =>    |-  ( ph  ->  ( G `  C )  =  ( y  e.  ( R1 `  C )  |->  if ( C  =  U. C ,  ( ( suc  U. ran  U. ( G " C )  .o  ( rank `  y )
 )  +o  ( ( G `  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  ( H " y ) ) ) ) )
 
Theoremdfac12lem2 8572* Lemma for dfac12 8577. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  F : ~P (har `  ( R1 `  A ) ) -1-1-> On )   &    |-  G  = recs (
 ( x  e.  _V  |->  ( y  e.  ( R1 `  dom  x ) 
 |->  if ( dom  x  =  U. dom  x ,  ( ( suc  U. ran  U.
 ran  x  .o  ( rank `  y ) )  +o  ( ( x `
  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  (
 ( `'OrdIso (  _E  ,  ran  ( x `  U. dom  x ) )  o.  ( x `  U. dom  x ) ) " y
 ) ) ) ) ) )   &    |-  ( ph  ->  C  e.  On )   &    |-  H  =  ( `'OrdIso (  _E  ,  ran  ( G `  U. C ) )  o.  ( G `  U. C ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  A. z  e.  C  ( G `  z ) : ( R1 `  z ) -1-1-> On )   =>    |-  ( ph  ->  ( G `  C ) : ( R1 `  C )
 -1-1-> On )
 
Theoremdfac12lem3 8573* Lemma for dfac12 8577. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  F : ~P (har `  ( R1 `  A ) ) -1-1-> On )   &    |-  G  = recs (
 ( x  e.  _V  |->  ( y  e.  ( R1 `  dom  x ) 
 |->  if ( dom  x  =  U. dom  x ,  ( ( suc  U. ran  U.
 ran  x  .o  ( rank `  y ) )  +o  ( ( x `
  suc  ( rank `  y ) ) `  y ) ) ,  ( F `  (
 ( `'OrdIso (  _E  ,  ran  ( x `  U. dom  x ) )  o.  ( x `  U. dom  x ) ) " y
 ) ) ) ) ) )   =>    |-  ( ph  ->  ( R1 `  A )  e. 
 dom  card )
 
Theoremdfac12r 8574 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 8577 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( A. x  e. 
 On  ~P x  e.  dom  card  <->  U. ( R1 " On )  C_  dom  card )
 
Theoremdfac12k 8575* Equivalence of dfac12 8577 and dfac12a 8576, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A. x  e. 
 On  ~P x  e.  dom  card  <->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
 
Theoremdfac12a 8576 The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  (CHOICE  <->  A. x  e.  On  ~P x  e.  dom  card )
 
Theoremdfac12 8577 The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
 
Theoremkmlem1 8578* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)
 |-  ( A. x ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ph )  ->  E. y A. z  e.  x  ps )  ->  A. x ( A. z  e.  x  A. w  e.  x  ph  ->  E. y A. z  e.  x  ( z  =/=  (/)  ->  ps ) ) )
 
Theoremkmlem2 8579* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  ( E. y A. z  e.  x  ( ph  ->  E! w  w  e.  ( z  i^i  y ) )  <->  E. y ( -.  y  e.  x  /\  A. z  e.  x  (
 ph  ->  E! w  w  e.  ( z  i^i  y ) ) ) )
 
Theoremkmlem3 8580* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.)
 |-  ( ( z  \  U. ( x  \  {
 z } ) )  =/=  (/)  <->  E. v  e.  z  A. w  e.  x  ( z  =/=  w  ->  -.  v  e.  (
 z  i^i  w )
 ) )
 
Theoremkmlem4 8581* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
 |-  ( ( w  e.  x  /\  z  =/= 
 w )  ->  (
 ( z  \  U. ( x  \  { z } ) )  i^i 
 w )  =  (/) )
 
Theoremkmlem5 8582* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  ( ( w  e.  x  /\  z  =/= 
 w )  ->  (
 ( z  \  U. ( x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) )  =  (/) )
 
Theoremkmlem6 8583* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
 |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  (
 ph  ->  A  =  (/) ) )  ->  A. z  e.  x  E. v  e.  z  A. w  e.  x  ( ph  ->  -.  v  e.  A ) )
 
Theoremkmlem7 8584* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
 |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  -.  E. z  e.  x  A. v  e.  z  E. w  e.  x  (
 z  =/=  w  /\  v  e.  ( z  i^i  w ) ) )
 
Theoremkmlem8 8585* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ( -.  E. z  e.  u  A. w  e.  z  ps  ->  E. y A. z  e.  u  ( z  =/= 
 (/)  ->  E! w  w  e.  ( z  i^i  y ) ) )  <-> 
 ( E. z  e.  u  A. w  e.  z  ps  \/  E. y ( -.  y  e.  u  /\  A. z  e.  u  E! w  w  e.  ( z  i^i  y ) ) ) )
 
Theoremkmlem9 8586* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |- 
 A. z  e.  A  A. w  e.  A  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )
 
Theoremkmlem10 8587* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( A. h (
 A. z  e.  h  A. w  e.  h  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) )  ->  E. y A. z  e.  h  ph )  ->  E. y A. z  e.  A  ph )
 
Theoremkmlem11 8588* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( z  e.  x  ->  ( z  i^i  U. A )  =  (
 z  \  U. ( x 
 \  { z }
 ) ) )
 
Theoremkmlem12 8589* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( A. z  e.  x  ( z  \  U. ( x  \  {
 z } ) )  =/=  (/)  ->  ( A. z  e.  A  (
 z  =/=  (/)  ->  E! v  v  e.  (
 z  i^i  y )
 )  ->  A. z  e.  x  ( z  =/=  (/)  ->  E! v  v  e.  ( z  i^i  ( y  i^i  U. A ) ) ) ) )
 
Theoremkmlem13 8590* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.)
 |-  A  =  { u  |  E. t  e.  x  u  =  ( t  \ 
 U. ( x  \  { t } )
 ) }   =>    |-  ( A. x ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  (
 z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )  <->  A. x ( -. 
 E. z  e.  x  A. v  e.  z  E. w  e.  x  (
 z  =/=  w  /\  v  e.  ( z  i^i  w ) )  ->  E. y A. z  e.  x  ( z  =/=  (/)  ->  E! v  v  e.  ( z  i^i  y ) ) ) )
 
Theoremkmlem14 8591* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ph  <->  ( z  e.  y  ->  ( (
 v  e.  x  /\  y  =/=  v )  /\  z  e.  v )
 ) )   &    |-  ( ps  <->  ( z  e.  x  ->  ( (
 v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )   &    |-  ( ch  <->  A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )   =>    |-  ( E. z  e.  x  A. v  e.  z  E. w  e.  x  ( z  =/= 
 w  /\  v  e.  ( z  i^i  w ) )  <->  E. y A. z E. v A. u ( y  e.  x  /\  ph ) )
 
Theoremkmlem15 8592* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ph  <->  ( z  e.  y  ->  ( (
 v  e.  x  /\  y  =/=  v )  /\  z  e.  v )
 ) )   &    |-  ( ps  <->  ( z  e.  x  ->  ( (
 v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )   &    |-  ( ch  <->  A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )   =>    |-  ( ( -.  y  e.  x  /\  ch )  <->  A. z E. v A. u ( -.  y  e.  x  /\  ps )
 )
 
Theoremkmlem16 8593* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
 |-  ( ph  <->  ( z  e.  y  ->  ( (
 v  e.  x  /\  y  =/=  v )  /\  z  e.  v )
 ) )   &    |-  ( ps  <->  ( z  e.  x  ->  ( (
 v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )   &    |-  ( ch  <->  A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )   =>    |-  ( ( E. z  e.  x  A. v  e.  z  E. w  e.  x  ( z  =/= 
 w  /\  v  e.  ( z  i^i  w ) )  \/  E. y
 ( -.  y  e.  x  /\  ch )
 ) 
 <-> 
 E. y A. z E. v A. u ( ( y  e.  x  /\  ph )  \/  ( -.  y  e.  x  /\  ps ) ) )
 
Theoremdfackm 8594* Equivalence of the Axiom of Choice and Maes' AC ackm 8893. The proof consists of lemmas kmlem1 8578 through kmlem16 8593 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 8557 with biid 239) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x E. y A. z E. v A. u ( ( y  e.  x  /\  ( z  e.  y  ->  (
 ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v ) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
 ( v  e.  z  /\  v  e.  y
 )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) ) ) )
 
2.6.9  Cardinal number arithmetic
 
Syntaxccda 8595 Extend class definition to include cardinal number addition.
 class  +c
 
Definitiondf-cda 8596* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 8598 for its value and a description. (Contributed by NM, 24-Sep-2004.)
 |- 
 +c  =  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  X.  { (/) } )  u.  ( y  X.  { 1o } ) ) )
 
Theoremcdafn 8597 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 +c  Fn  ( _V  X. 
 _V )
 
Theoremcdaval 8598 Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while Cartesian product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 8974, carddom 8977, and cardsdom 8978. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B )  =  (
 ( A  X.  { (/)
 } )  u.  ( B  X.  { 1o }
 ) ) )
 
Theoremuncdadom 8599 Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
 
Theoremcdaun 8600 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A  +c  B ) 
 ~~  ( A  u.  B ) )
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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 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300
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