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Theorem List for Metamath Proof Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitunitc 8001* The union of all union iterates creates the transitive closure; compare trcl 7364. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( TC `  A )  =  U. ran  (  U `  A )
 
Theoremituniiun 8002* Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( A  e.  V  ->  ( ( U `  A ) `  suc  B )  =  U_ a  e.  A  ( ( U `
  a ) `  B ) )
 
Theoremhsmexlem7 8003* Lemma for hsmex 8012. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  H  =  ( rec ( ( z  e. 
 _V  |->  (har `  ~P ( X  X.  z
 ) ) ) ,  (har `  ~P X ) )  |`  om )   =>    |-  ( H `  (/) )  =  (har `  ~P X )
 
Theoremhsmexlem8 8004* Lemma for hsmex 8012. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  H  =  ( rec ( ( z  e. 
 _V  |->  (har `  ~P ( X  X.  z
 ) ) ) ,  (har `  ~P X ) )  |`  om )   =>    |-  (
 a  e.  om  ->  ( H `  suc  a
 )  =  (har `  ~P ( X  X.  ( H `  a ) ) ) )
 
Theoremhsmexlem9 8005* Lemma for hsmex 8012. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  H  =  ( rec ( ( z  e. 
 _V  |->  (har `  ~P ( X  X.  z
 ) ) ) ,  (har `  ~P X ) )  |`  om )   =>    |-  (
 a  e.  om  ->  ( H `  a )  e.  On )
 
Theoremhsmexlem1 8006 Lemma for hsmex 8012. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  O  = OrdIso (  _E 
 ,  A )   =>    |-  ( ( A 
 C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  (har `  ~P B ) )
 
Theoremhsmexlem2 8007* Lemma for hsmex 8012. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8151 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  F  = OrdIso (  _E 
 ,  B )   &    |-  G  = OrdIso (  _E  ,  U_ a  e.  A  B )   =>    |-  ( ( A  e.  _V 
 /\  C  e.  On  /\ 
 A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  dom  G  e.  (har `  ~P ( A  X.  C ) ) )
 
Theoremhsmexlem3 8008* Lemma for hsmex 8012. Clear  I hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  F  = OrdIso (  _E 
 ,  B )   &    |-  G  = OrdIso (  _E  ,  U_ a  e.  A  B )   =>    |-  ( ( ( A  ~<_*  D 
 /\  C  e.  On )  /\  A. a  e.  A  ( B  e.  ~P On  /\  dom  F  e.  C ) )  ->  dom  G  e.  (har `  ~P ( D  X.  C ) ) )
 
Theoremhsmexlem4 8009* Lemma for hsmex 8012. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  X  e.  _V   &    |-  H  =  ( rec ( ( z  e.  _V  |->  (har `  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )   &    |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
 ) ,  x )  |`  om ) )   &    |-  S  =  { a  e.  U. ( R1 " On )  |  A. b  e.  ( TC `  { a }
 ) b  ~<_  X }   &    |-  O  = OrdIso (  _E  ,  ( rank " ( ( U `
  d ) `  c ) ) )   =>    |-  ( ( c  e. 
 om  /\  d  e.  S )  ->  dom  O  e.  ( H `  c
 ) )
 
Theoremhsmexlem5 8010* Lemma for hsmex 8012. Combining the above constraints, along with itunitc 8001 and tcrank 7508, gives an effective constraint on the rank of  S. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  X  e.  _V   &    |-  H  =  ( rec ( ( z  e.  _V  |->  (har `  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )   &    |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
 ) ,  x )  |`  om ) )   &    |-  S  =  { a  e.  U. ( R1 " On )  |  A. b  e.  ( TC `  { a }
 ) b  ~<_  X }   &    |-  O  = OrdIso (  _E  ,  ( rank " ( ( U `
  d ) `  c ) ) )   =>    |-  ( d  e.  S  ->  ( rank `  d )  e.  (har `  ~P ( om  X. 
 U. ran  H )
 ) )
 
Theoremhsmexlem6 8011* Lemmr for hsmex 8012. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  X  e.  _V   &    |-  H  =  ( rec ( ( z  e.  _V  |->  (har `  ~P ( X  X.  z ) ) ) ,  (har `  ~P X ) )  |`  om )   &    |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e.  _V  |->  U. y
 ) ,  x )  |`  om ) )   &    |-  S  =  { a  e.  U. ( R1 " On )  |  A. b  e.  ( TC `  { a }
 ) b  ~<_  X }   &    |-  O  = OrdIso (  _E  ,  ( rank " ( ( U `
  d ) `  c ) ) )   =>    |-  S  e.  _V
 
Theoremhsmex 8012* The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 7260. (Contributed by Stefan O'Rear, 14-Feb-2015.)
 |-  ( X  e.  V  ->  { s  e.  U. ( R1 " On )  |  A. x  e.  ( TC `  { s }
 ) x  ~<_  X }  e.  _V )
 
Theoremhsmex2 8013* The set of hereditary size-limited sets, assuming ax-reg 7260. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  { s  |  A. x  e.  ( TC ` 
 { s } ) x 
 ~<_  X }  e.  _V )
 
Theoremhsmex3 8014* The set of hereditary size-limited sets, assuming ax-reg 7260, using strict comparison (an easy corrolary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  { s  |  A. x  e.  ( TC ` 
 { s } ) x  ~<  X }  e.  _V )
 
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY

In this section we add the Axiom of Choice ax-ac 8039, as well as weaker forms such as the axiom of countable choice ax-cc 8015 and dependent choice ax-dc 8026. We introduce these weaker forms so that theorems that do not need the full power of the axiom of choice, but need more than simple ZF, can use these intermediate axioms instead.

The combination of the Zermel-Fraenkel axioms and the axiom of choice is often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics.

However, there have been and still are some lingering controversies about the Axiom of Choice. The axiom of choice does not satisfy those who wish to have a constructive proof (e.g., it will not satify intuitionist logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms.

 
3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
 
Axiomax-cc 8015* The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8058, but is weak enough that it can be proven using DC (see axcc 8038). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of non-empty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.)
 |-  ( x  ~~  om  ->  E. f A. z  e.  x  ( z  =/= 
 (/)  ->  ( f `  z )  e.  z
 ) )
 
Theoremaxcc2lem 8016* Lemma for axcc2 8017. (Contributed by Mario Carneiro, 8-Feb-2013.)
 |-  K  =  ( n  e.  om  |->  if (
 ( F `  n )  =  (/) ,  { (/)
 } ,  ( F `
  n ) ) )   &    |-  A  =  ( n  e.  om  |->  ( { n }  X.  ( K `  n ) ) )   &    |-  G  =  ( n  e.  om  |->  ( 2nd `  ( f `  ( A `  n ) ) ) )   =>    |-  E. g ( g  Fn 
 om  /\  A. n  e. 
 om  ( ( F `
  n )  =/=  (/)  ->  ( g `  n )  e.  ( F `  n ) ) )
 
Theoremaxcc2 8017* A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
 |- 
 E. g ( g  Fn  om  /\  A. n  e.  om  ( ( F `  n )  =/=  (/)  ->  ( g `  n )  e.  ( F `  n ) ) )
 
Theoremaxcc3 8018* A possibly more useful version of ax-cc 8015 using sequences  F
( n ) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  F  e.  _V   &    |-  N  ~~ 
 om   =>    |- 
 E. f ( f  Fn  N  /\  A. n  e.  N  ( F  =/=  (/)  ->  ( f `  n )  e.  F ) )
 
Theoremaxcc4 8019* A version of axcc3 8018 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
 |-  A  e.  _V   &    |-  N  ~~ 
 om   &    |-  ( x  =  ( f `  n ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. n  e.  N  E. x  e.  A  ph  ->  E. f
 ( f : N --> A  /\  A. n  e.  N  ps ) )
 
Theoremacncc 8020 An ax-cc 8015 equivalent: every set has choice sets of length  om. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |- AC  om  =  _V
 
Theoremaxcc4dom 8021* Relax the constraint on axcc4 8019 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.)
 |-  A  e.  _V   &    |-  ( x  =  ( f `  n )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( N  ~<_ 
 om  /\  A. n  e.  N  E. x  e.  A  ph )  ->  E. f ( f : N --> A  /\  A. n  e.  N  ps ) )
 
Theoremdomtriomlem 8022* Lemma for domtriom 8023. (Contributed by Mario Carneiro, 9-Feb-2013.)
 |-  A  e.  _V   &    |-  B  =  { y  |  ( y  C_  A  /\  y  ~~  ~P n ) }   &    |-  C  =  ( n  e.  om  |->  ( ( b `  n )  \  U_ k  e.  n  ( b `  k ) ) )   =>    |-  ( -.  A  e.  Fin  ->  om 
 ~<_  A )
 
Theoremdomtriom 8023 Trichotomy of equinumerosity for 
om, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 7894) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( om  ~<_  A  <->  -.  A  ~<  om )
 
Theoremfin41 8024 Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.)
 |- FinIV  = 
 Fin
 
Theoremdominf 8025 A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8015. See dominfac 8149 for a version proved from ax-ac 8039. The axiom of Regularity is used for this proof, via inf3lem6 7288, and its use is necessary: otherwise the set  A  =  { A } or  A  =  { (/)
,  A } (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A  C_  U. A )  ->  om  ~<_  A )
 
Axiomax-dc 8026* Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 8102. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  ( ( E. y E. z  y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n ) )
 
Theoremdcomex 8027 The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
 |- 
 om  e.  _V
 
Theoremaxdc2lem 8028* Lemma for axdc2 8029. We construct a relation  R based on  F such that  x R y iff  y  e.  ( F `
 x ), and show that the "function" described by ax-dc 8026 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  A  e.  _V   &    |-  R  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }   &    |-  G  =  ( x  e.  om  |->  ( h `  x ) )   =>    |-  ( ( A  =/=  (/)  /\  F : A --> ( ~P A  \  { (/) } )
 )  ->  E. g
 ( g : om --> A  /\  A. k  e. 
 om  ( g `  suc  k )  e.  ( F `  ( g `  k ) ) ) )
 
Theoremaxdc2 8029* An apparent strengthening of ax-dc 8026 (but derived from it) which shows that there is a denumerable sequence  g for any function that maps elements of a set  A to nonempty subsets of 
A such that  g (
x  +  1 )  e.  F ( g ( x ) ) for all  x  e.  om. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  F : A
 --> ( ~P A  \  { (/) } ) ) 
 ->  E. g ( g : om --> A  /\  A. k  e.  om  (
 g `  suc  k )  e.  ( F `  ( g `  k
 ) ) ) )
 
Theoremaxdc3lem 8030* The class  S of finite approximations to the DC sequence is a set. (We derive here the stronger statement that  S is a subset of a specific set, namely  ~P ( om  X.  A ).) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.)
 |-  A  e.  _V   &    |-  S  =  { s  |  E. n  e.  om  ( s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
 s `  k )
 ) ) }   =>    |-  S  e.  _V
 
Theoremaxdc3lem2 8031* Lemma for axdc3 8034. We have constructed a "candidate set"  S, which consists of all finite sequences  s that satisfy our property of interest, namely  s ( x  + 
1 )  e.  F
( s ( x ) ) on its domain, but with the added constraint that 
s ( 0 )  =  C. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 8026 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely  ( h `  n ) : m --> A (for some integer  m). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 8026 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence  h, we can construct the sequence  g that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
 |-  A  e.  _V   &    |-  S  =  { s  |  E. n  e.  om  ( s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
 s `  k )
 ) ) }   &    |-  G  =  ( x  e.  S  |->  { y  e.  S  |  ( dom  y  =  suc  dom 
 x  /\  ( y  |` 
 dom  x )  =  x ) } )   =>    |-  ( E. h ( h : om
 --> S  /\  A. k  e.  om  ( h `  suc  k )  e.  ( G `  ( h `  k ) ) ) 
 ->  E. g ( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  ( g `  suc  k )  e.  ( F `  ( g `  k ) ) ) )
 
Theoremaxdc3lem3 8032* Simple substitution lemma for axdc3 8034. (Contributed by Mario Carneiro, 27-Jan-2013.)
 |-  A  e.  _V   &    |-  S  =  { s  |  E. n  e.  om  ( s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
 s `  k )
 ) ) }   &    |-  B  e.  _V   =>    |-  ( B  e.  S  <->  E. m  e.  om  ( B : suc  m --> A  /\  ( B `  (/) )  =  C  /\  A. k  e.  m  ( B ` 
 suc  k )  e.  ( F `  ( B `  k ) ) ) )
 
Theoremaxdc3lem4 8033* Lemma for axdc3 8034. We have constructed a "candidate set"  S, which consists of all finite sequences  s that satisfy our property of interest, namely  s ( x  + 
1 )  e.  F
( s ( x ) ) on its domain, but with the added constraint that 
s ( 0 )  =  C. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 8026 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely  ( h `  n ) : m --> A (for some integer  m). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 8026 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that  S is nonempty, and that  G always maps to a nonempty subset of  S, so that we can apply axdc2 8029. See axdc3lem2 8031 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.)
 |-  A  e.  _V   &    |-  S  =  { s  |  E. n  e.  om  ( s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
 s `  k )
 ) ) }   &    |-  G  =  ( x  e.  S  |->  { y  e.  S  |  ( dom  y  =  suc  dom 
 x  /\  ( y  |` 
 dom  x )  =  x ) } )   =>    |-  (
 ( C  e.  A  /\  F : A --> ( ~P A  \  { (/) } )
 )  ->  E. g
 ( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  (
 g `  suc  k )  e.  ( F `  ( g `  k
 ) ) ) )
 
Theoremaxdc3 8034* Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value  C. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.)
 |-  A  e.  _V   =>    |-  ( ( C  e.  A  /\  F : A --> ( ~P A  \  { (/) } ) ) 
 ->  E. g ( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  ( g `  suc  k )  e.  ( F `  ( g `  k ) ) ) )
 
Theoremaxdc4lem 8035* Lemma for axdc4 8036. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |-  A  e.  _V   &    |-  G  =  ( n  e.  om ,  x  e.  A  |->  ( { suc  n }  X.  ( n F x ) ) )   =>    |-  ( ( C  e.  A  /\  F : ( om  X.  A ) --> ( ~P A  \  { (/) } )
 )  ->  E. g
 ( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  (
 g `  suc  k )  e.  ( k F ( g `  k
 ) ) ) )
 
Theoremaxdc4 8036* A more general version of axdc3 8034 that allows the function  F to vary with  k. (Contributed by Mario Carneiro, 31-Jan-2013.)
 |-  A  e.  _V   =>    |-  ( ( C  e.  A  /\  F : ( om  X.  A ) --> ( ~P A  \  { (/) } )
 )  ->  E. g
 ( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  (
 g `  suc  k )  e.  ( k F ( g `  k
 ) ) ) )
 
Theoremaxcclem 8037* Lemma for axcc 8038. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |-  A  =  ( x 
 \  { (/) } )   &    |-  F  =  ( n  e.  om ,  y  e.  U. A  |->  ( f `  n ) )   &    |-  G  =  ( w  e.  A  |->  ( h `  suc  ( `' f `  w ) ) )   =>    |-  ( x  ~~  om  ->  E. g A. z  e.  x  ( z  =/= 
 (/)  ->  ( g `  z )  e.  z
 ) )
 
Theoremaxcc 8038* Although CC can be proven trivially using ac5 8058, we prove it here using DC. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  ( x  ~~  om  ->  E. f A. z  e.  x  ( z  =/= 
 (/)  ->  ( f `  z )  e.  z
 ) )
 
3.2  ZFC Set Theory - add the Axiom of Choice
 
3.2.1  Introduce the Axiom of Choice
 
Axiomax-ac 8039* Axiom of Choice. The Axiom of Choice (AC) is usually considered an extension of ZF set theory rather than a proper part of it. It is sometimes considered philosophically controversial because it asserts the existence of a set without telling us what the set is. ZF set theory that includes AC is called ZFC.

The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry in the pair is the member of  x, and the other entry is some arbitrary member of that member of  x. See the rewritten version ac3 8042 for a more detailed explanation. Theorem ac2 8041 shows an equivalent written compactly with restricted quantifiers.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 8046 is slightly shorter when the biconditional of ax-ac 8039 is expanded into implication and negation. In axac3 8044 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8249 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8073, ac5 8058, and ac7 8054. The Axiom of Regularity ax-reg 7260 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7711. Equivalents to AC are the well-ordering theorem weth 8076 and Zorn's lemma zorn 8088. See ac4 8056 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7260 for derivation of AC equivalents, we provide ax-ac2 8043 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8043 from ax-ac 8039 is shown by theorem axac2 8047, and the reverse derivation by axac 8048. (Contributed by NM, 18-Jul-1996.)

 |- 
 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 ) )
 
Theoremzfac 8040* Axiom of Choice expressed with fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8039. (Contributed by NM, 14-Aug-2003.)
 |- 
 E. x A. y A. z ( ( y  e.  z  /\  z  e.  w )  ->  E. w A. y ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  (
 y  e.  w  /\  w  e.  x )
 ) 
 <->  y  =  w ) )
 
Theoremac2 8041* Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 8042 is easier to understand.) Note: aceq0 7699 shows the logical equivalence to ax-ac 8039. (Contributed by NM, 18-Jul-1996.)
 |- 
 E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  ( z  e.  u  /\  v  e.  u )
 
Theoremac3 8042* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8039 can be established by chaining aceq0 7699 and aceq2 7700. A standard textbook version of AC is derived from this one in dfac2a 7710, and this version of AC is derived from the textbook version in dfac2 7711.

The following sketch will help you understand this version of the axiom. Given any set  x, the axiom says that there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry in the pair is the member of 
x, and the other entry is some arbitrary member of that member of  x. Using the Axiom of Regularity, we can show that  y is really a set of ordered pairs, very similar to the ordered pair construction opthreg 7273. The key theorem for this (used in the proof of dfac2 7711) is preleq 7272. With this modified definition of ordered pair, it can be seen that  y is actually a choice function on the members of  x.

For example, suppose  x  =  { { 1 ,  2 } ,  { 1 ,  3 } ,  { 2 ,  3 ,  4 } }. Let us try  y  =  { { { 1 ,  2 } ,  1 } ,  { { 1 ,  3 } , 
1 } ,  { { 2 ,  3 ,  4 } ,  2 } }. For the member (of  x)  z  =  {
1 ,  2 }, the only assignment to  w and  v that satisfies the axiom is  w  =  1 and  v  =  { { 1 ,  2 } , 
1 }, so there is exactly one  w as required. We verify the other two members of  x similarly. Thus  y satisfies the axiom. Using our modified ordered pair definition, we can say that  y corresponds to the choice function  { <. { 1 ,  2 } , 
1 >. ,  <. { 1 ,  3 } , 
1 >. ,  <. { 2 ,  3 ,  4 } ,  2 >. }. Of course other choices for  y will also satisfy the axiom, for example  y  =  { { { 1 ,  2 } ,  2 } ,  { { 1 ,  3 } , 
1 } ,  { { 2 ,  3 ,  4 } ,  4 } }. What AC tells us is that there exists at least one such  y, but it doesn't tell us which one. (Contributed by NM, 19-Jul-1996.)

 |- 
 E. y A. z  e.  x  ( z  =/= 
 (/)  ->  E! w  e.  z  E. v  e.  y  ( z  e.  v  /\  w  e.  v ) )
 
Axiomax-ac2 8043* In order to avoid uses of ax-reg 7260 for derivation of AC equivalents, we provide ax-ac2 8043, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as theorem ackm 8046. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1536 available. The derivation of ax-ac2 8043 from ax-ac 8039 is shown by theorem axac2 8047, and the reverse derivation by axac 8048. Note that we use ax-reg 7260 to derive ax-ac 8039 from ax-ac2 8043, but not to derive ax-ac2 8043 from ax-ac 8039. (Contributed by NM, 19-Dec-2016.)
 |- 
 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 ) ) ) ) )
 
Theoremaxac3 8044 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8043 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
 |- CHOICE
 
Theoremaxac3OLD 8045 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac 8039 as an axiom. Obsolete as of 20-Dec-2016. (Contributed by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
 |- CHOICE
 
Theoremackm 8046* A remarkable equivalent to the Axiom of Choice that has only 5 quantifiers (when expanded to 
e.,  = primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 7746. Maes found this version of AC in April, 2004 (replacing a longer version, also with 5 quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier ( e.,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

 |- 
 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 ) ) ) ) )
 
Theoremaxac2 8047* Derive ax-ac2 8043 from ax-ac 8039. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
 |- 
 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 ) ) ) ) )
 
Theoremaxac 8048* Derive ax-ac 8039 from ax-ac2 8043. Note that ax-reg 7260 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
 |- 
 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 ) )
 
Theoremaxaci 8049 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x ph )   =>    |-  ph
 
Theoremcardeqv 8050 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 dom  card  =  _V
 
Theoremnumth3 8051 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  dom  card )
 
Theoremnumth2 8052* Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)
 |-  A  e.  _V   =>    |-  E. x  e. 
 On  x  ~~  A
 
Theoremnumth 8053* Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
 |-  A  e.  _V   =>    |-  E. x  e. 
 On  E. f  f : x -1-1-onto-> A
 
Theoremac7 8054* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
 |- 
 E. f ( f 
 C_  x  /\  f  Fn  dom  x )
 
Theoremac7g 8055* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
 |-  ( R  e.  A  ->  E. f ( f 
 C_  R  /\  f  Fn  dom  R ) )
 
Theoremac4 8056* Equivalent of Axiom of Choice. We do not insist that  f be a function. However, theorem ac5 8058, derived from this one, shows that this form of the axiom does imply that at least one such set  f whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice"  E. F A. z
( z  =/=  (/)  ->  ( F `  z )  e.  z ), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable  F and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 8072. (Contributed by NM, 21-Jul-1996.)

 |- 
 E. f A. z  e.  x  ( z  =/= 
 (/)  ->  ( f `  z )  e.  z
 )
 
Theoremac4c 8057* Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.)
 |-  A  e.  _V   =>    |-  E. f A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x )
 
Theoremac5 8058* An Axiom of Choice equivalent: there exists a function  f (called a choice function) with domain 
A that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that  f be a function is not necessary; see ac4 8056. (Contributed by NM, 29-Aug-1999.)
 |-  A  e.  _V   =>    |-  E. f ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
 f `  x )  e.  x ) )
 
Theoremac5b 8059* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  x  =/=  (/)  ->  E. f
 ( f : A --> U. A  /\  A. x  e.  A  ( f `  x )  e.  x ) )
 
Theoremac6num 8060* A version of ac6 8061 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  V  /\  U_ x  e.  A  { y  e.  B  |  ph }  e.  dom  card  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6 8061* Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set  B, where  ph depends on  x (the natural number) and  y (to specify a member of  B). A stronger version of this theorem, ac6s 8065, allows  B to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( y  =  ( f `  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6c4 8062* Equivalent of Axiom of Choice.  B is a collection  B ( x ) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )
 )
 
Theoremac6c5 8063* Equivalent of Axiom of Choice.  B is a collection  B ( x ) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f A. x  e.  A  ( f `  x )  e.  B )
 
Theoremac9 8064* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
 
Theoremac6s 8065* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 7517, we derive this strong version of ac6 8061 that doesn't require  B to be a set. (Contributed by NM, 4-Feb-2004.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6n 8066* Equivalent of Axiom of Choice. Contrapositive of ac6s 8065. (Contributed by NM, 10-Jun-2007.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. f
 ( f : A --> B  ->  E. x  e.  A  ps )  ->  E. x  e.  A  A. y  e.  B  ph )
 
Theoremac6s2 8067* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8068. (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y ph  ->  E. f ( f  Fn  A  /\  A. x  e.  A  ps ) )
 
Theoremac6s3 8068* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y ph  ->  E. f A. x  e.  A  ps )
 
Theoremac6sg 8069* ac6s 8065 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) ) )
 
Theoremac6sf 8070* Version of ac6 8061 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
 |- 
 F/ y ps   &    |-  A  e.  _V   &    |-  ( y  =  ( f `  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6s4 8071* Generalization of the Axiom of Choice to proper classes.  B is a collection  B ( x ) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )
 )
 
Theoremac6s5 8072* Generalization of the Axiom of Choice to proper classes.  B is a collection  B ( x ) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f A. x  e.  A  ( f `  x )  e.  B )
 
Theoremac8 8073* An Axiom of Choice equivalent. Given a family  x of mutually disjoint nonempty sets, there exists a set  y containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
 |-  ( ( 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 ) )
 
Theoremac9s 8074* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes  B ( x ) (achieved via the Collection Principle cp 7515). (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
 
3.2.2  AC equivalents: well ordering, Zorn's lemma
 
Theoremnumthcor 8075* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x )
 
Theoremweth 8076* Well-ordering theorem: any set  A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  V  ->  E. x  x  We  A )
 
Theoremzorn2lem1 8077* Lemma for zorn2 8087. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) )  ->  ( F `  x )  e.  D )
 
Theoremzorn2lem2 8078* Lemma for zorn2 8087. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) )  ->  ( y  e.  x  ->  ( F `  y
 ) R ( F `
  x ) ) )
 
Theoremzorn2lem3 8079* Lemma for zorn2 8087. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) ) ) 
 ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
 
Theoremzorn2lem4 8080* Lemma for zorn2 8087. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( R  Po  A  /\  w  We  A )  ->  E. x  e.  On  D  =  (/) )
 
Theoremzorn2lem5 8081* Lemma for zorn2 8087. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  ->  ( F " x )  C_  A )
 
Theoremzorn2lem6 8082* Lemma for zorn2 8087. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( R  Po  A  ->  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  ->  R  Or  ( F " x ) ) )
 
Theoremzorn2lem7 8083* Lemma for zorn2 8087. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. s
 ( ( s  C_  A  /\  R  Or  s
 )  ->  E. a  e.  A  A. r  e.  s  ( r R a  \/  r  =  a ) ) ) 
 ->  E. a  e.  A  A. b  e.  A  -.  a R b )
 
Theoremzorn2g 8084* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8087 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. w ( ( w  C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x )
 ) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremzorng 8085* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 8088 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  A. z ( ( z  C_  A  /\ [ C.]  Or  z ) 
 ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzornn0g 8086* Variant of Zorn's lemma zorng 8085 in which  (/), the union of the empty chain, is not required to be an element of  A. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  A  =/=  (/)  /\  A. z ( ( z 
 C_  A  /\  z  =/= 
 (/)  /\ [ C.]  Or  z
 )  ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzorn2 8087* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set  A (with an ordering relation  R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 8077 through zorn2lem7 8083; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8083. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( R  Po  A  /\  A. w ( ( w 
 C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x ) ) ) 
 ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremzorn 8088* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 8087 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
 |-  A  e.  _V   =>    |-  ( A. z
 ( ( z  C_  A  /\ [ C.]  Or  z
 )  ->  U. z  e.  A )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzornn0 8089* Variant of Zorn's lemma zorn 8088 in which  (/), the union of the empty chain, is not required to be an element of  A. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A. z ( ( z  C_  A  /\  z  =/=  (/)  /\ [ C.]  Or  z )  ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremttukeylem1 8090* Lemma for ttukey 8099. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   =>    |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
 
Theoremttukeylem2 8091* Lemma for ttukey 8099. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   =>    |-  ( ( ph  /\  ( C  e.  A  /\  D  C_  C ) ) 
 ->  D  e.  A )
 
Theoremttukeylem3 8092* Lemma for ttukey 8099. (Contributed by Mario Carneiro, 11-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  C  e.  On )  ->  ( G `  C )  =  if ( C  =  U. C ,  if ( C  =  (/) ,  B ,  U. ( G " C ) ) ,  ( ( G `  U. C )  u.  if ( ( ( G `
  U. C )  u. 
 { ( F `  U. C ) } )  e.  A ,  { ( F `  U. C ) } ,  (/) ) ) ) )
 
Theoremttukeylem4 8093* Lemma for ttukey 8099. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ph  ->  ( G `  (/) )  =  B )
 
Theoremttukeylem5 8094* Lemma for ttukey 8099. The  G function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  ( C  e.  On  /\  D  e.  On  /\  C  C_  D ) )  ->  ( G `  C ) 
 C_  ( G `  D ) )
 
Theoremttukeylem6 8095* Lemma for ttukey 8099. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  C  e.  suc  ( card `  ( U. A  \  B ) ) )  ->  ( G `  C )  e.  A )
 
Theoremttukeylem7 8096* Lemma for ttukey 8099. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ph  ->  E. x  e.  A  ( B  C_  x  /\  A. y  e.  A  -.  x  C.  y ) )
 
Theoremttukey2g 8097* The Teichmüller-Tukey Lemma ttukey 8099 with a slightly stronger conclusion: we can set up the maximal element of  A so that it also contains some given  B  e.  A as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( U. A  e.  dom  card  /\  B  e.  A  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  ( B  C_  x  /\  A. y  e.  A  -.  x  C.  y ) )
 
Theoremttukeyg 8098* The Teichmüller-Tukey Lemma ttukey 8099 stated with the "choice" as an antecedent (the hypothesis  U. A  e.  dom  card says that  U. A is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( U. A  e.  dom  card  /\  A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremttukey 8099* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If  A is a nonempty collection of finite character, then  A has a maximal element with respect to inclusion. Here "finite character" means that  x  e.  A iff every finite subset of  x is in  A. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremaxdclem 8100* Lemma for axdc 8102. (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  F  =  ( rec ( ( y  e. 
 _V  |->  ( g `  { z  |  y x z } )
 ) ,  s )  |`  om )   =>    |-  ( ( A. y  e.  ~P  dom  x (
 y  =/=  (/)  ->  (
 g `  y )  e.  y )  /\  ran  x 
 C_  dom  x  /\  E. z ( F `  K ) x z )  ->  ( K  e.  om  ->  ( F `  K ) x ( F `  suc  K ) ) )
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