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Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalephordi 7701 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 7702 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 7703 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 7704 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 7705 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 7706 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 7707* An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7259 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 7573. (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 7708 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 7709 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 7710 Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  On  <->  Lim  ( aleph `  A )
 )
 
Theoremaleph11 7711 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 7712 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 7730. (Contributed by by Mario Carneiro, 2-Feb-2013.)
 |-  aleph : On -1-1-> On
 
Theoremalephsdom 7713 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 7714 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 7715 The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 7736, 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 7716* 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 7717* 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 7718* 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 7719 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 7720 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 7721 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 7722* 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 7723 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 7724* 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 7725 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 7726 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 7727 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 7728 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 7729 The aleph function is strictly monotone. (Contributed by by Mario Carneiro, 15-Mar-2013.)
 |- 
 Smo  aleph
 
Theoremalephf1ALT 7730 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. (Contributed by by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  aleph : On -1-1-> On
 
Theoremalephfplem1 7731 Lemma for alephfp 7735. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( H `  (/) )  e. 
 ran  aleph
 
Theoremalephfplem2 7732* Lemma for alephfp 7735. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w ) ) )
 
Theoremalephfplem3 7733* Lemma for alephfp 7735. (Contributed by NM, 6-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |-  ( v  e.  om  ->  ( H `  v
 )  e.  ran  aleph )
 
Theoremalephfplem4 7734 Lemma for alephfp 7735. (Contributed by NM, 5-Nov-2004.)
 |-  H  =  ( rec ( aleph ,  om )  |` 
 om )   =>    |- 
 U. ( H " om )  e.  ran  aleph
 
Theoremalephfp 7735 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 7736 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 7736 The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7735 for an actual example of a fixed point. Compare the inequality alephle 7715 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 7737* 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 7738 The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8302 or gchaleph2 8298.) The transposed form alephsucpw 8192 cannot be proven without the AC, and is in fact equlvalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |- 
 -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
 
Theoremmappwen 7739 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 7740* 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 7741 Countability of a countable union of finite sets with an 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 7742 Wff for an abbreviation of the axiom of choice.
 wff CHOICE
 
Definitiondf-ac 7743* 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 8085 as our definition, because the equivalence to more standard forms (dfac2 7757) 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 8085 itself as dfac0 7759. (Contributed by Mario Carneiro, 22-Feb-2015.)

 |-  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
 )
 
Theoremaceq1 7744* Equivalence of two versions of the Axiom of Choice ax-ac 8085. 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 7745* 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 8085. (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 7746* 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 7747* Lemma for dfac3 7748. (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 7748* Equivalence of two versions of the Axiom of Choice. The left-hand side is 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 7749* 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 7750* Lemma for dfac5 7755. (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 7751* Lemma for dfac5 7755. (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 7752* Lemma for dfac5 7755. (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 7753* Lemma for dfac5 7755. (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 7754* Lemma for dfac5 7755. (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 7755* 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 7756* Our Axiom of Choice (in the form of ac3 8088) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 7757 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 7757* Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 8088). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 7311 and preleq 7318 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 7756.) 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 7758* Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 8087). 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 7759* 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 8085. (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 7760* Equivalence of two versions of the Axiom of Choice ax-ac 8085. 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 7761* A proof of the equivalency of the Well Ordering Theorem weth 8122 and the Axiom of Choice ac7 8100. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  (CHOICE  <->  A. x E. r  r  We  x )
 
Theoremdfac9 7762* Equivalence of the axiom of choice with a statement related to ac9 8110; 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 7763 Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  (CHOICE  <->  dom  card  =  _V )
 
Theoremdfac10c 7764* 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 7765 Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 7743). (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  (CHOICE  <->  ( 
 ~~  " On )  =  _V )
 
Theoremacacni 7766 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 7767 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  (CHOICE  <->  A. xAC  x  =  _V )
 
Theoremdfac13 7768 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 7769* Lemma for dfac12 7775. (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 7770* Lemma for dfac12 7775. (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 7771* Lemma for dfac12 7775. (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 7772 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 7775 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 7773* Equivalence of dfac12 7775 and dfac12a 7774, 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 7774 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 7775 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 7776* 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 7777* 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 7778* 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 7779* 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 7780* 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 7781* 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 7782* 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 7783* 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 7784* 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 7785* 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 7786* 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 7787* 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 7788* 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 7789* 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 7790* 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 7791* 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 7792* Equivalence of the Axiom of Choice and Maes' AC ackm 8092. The proof consists of lemmas kmlem1 7776 through kmlem16 7791 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 7755 with biid 227) 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 7793 Extend class definition to include cardinal number addition.
 class  +c
 
Definitiondf-cda 7794* Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 7796 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 7795 Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 +c  Fn  ( _V  X. 
 _V )
 
Theoremcdaval 7796 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 cross 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 8173, carddom 8176, and cardsdom 8177. 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 7797 Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
 
Theoremcdaun 7798 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 ) )
 
Theoremcdaen 7799 Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  +c  C )  ~~  ( B  +c  D ) )
 
Theoremcdaenun 7800 Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  +c  C ) 
 ~~  ( B  u.  D ) )
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