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Theorem List for Metamath Proof Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisf32lem7 8001* Lemma for isfin3-2 8009. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   =>    |-  ( ( (
 ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  ( ( K `  A )  i^i  ( K `
  B ) )  =  (/) )
 
Theoremisf32lem8 8002* Lemma for isfin3-2 8009. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   =>    |-  ( ( ph  /\  A  e.  om )  ->  ( K `  A )  C_  G )
 
Theoremisf32lem9 8003* Lemma for isfin3-2 8009. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   &    |-  L  =  ( t  e.  G  |->  ( iota s ( s  e.  om  /\  t  e.  ( K `  s
 ) ) ) )   =>    |-  ( ph  ->  L : G -onto-> om )
 
Theoremisf32lem10 8004* Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ph  ->  F : om --> ~P G )   &    |-  ( ph  ->  A. x  e.  om  ( F `  suc  x )  C_  ( F `  x ) )   &    |-  ( ph  ->  -.  |^| ran  F  e.  ran  F )   &    |-  S  =  { y  e.  om  |  ( F `  suc  y )  C.  ( F `
  y ) }   &    |-  J  =  ( u  e.  om  |->  ( iota_ v  e.  S ( v  i^i  S ) 
 ~~  u ) )   &    |-  K  =  ( ( w  e.  S  |->  ( ( F `  w ) 
 \  ( F `  suc  w ) ) )  o.  J )   &    |-  L  =  ( t  e.  G  |->  ( iota s ( s  e.  om  /\  t  e.  ( K `  s
 ) ) ) )   =>    |-  ( ph  ->  ( G  e.  V  ->  om  ~<_*  G ) )
 
Theoremisf32lem11 8005* Lemma for isfin3-2 8009. Remove hypotheses from isf32lem10 8004. (Contributed by Stefan O'Rear, 17-May-2015.)
 |-  ( ( G  e.  V  /\  ( F : om
 --> ~P G  /\  A. b  e.  om  ( F `
  suc  b )  C_  ( F `  b
 )  /\  -.  |^| ran  F  e.  ran  F )
 )  ->  om  ~<_*  G )
 
Theoremisf32lem12 8006* Lemma for isfin3-2 8009. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om )
 ( A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }   =>    |-  ( G  e.  V  ->  ( -.  om  ~<_*  G  ->  G  e.  F ) )
 
Theoremisfin32i 8007 One half of isfin3-2 8009. (Contributed by Mario Carneiro, 3-Jun-2015.)
 |-  ( A  e. FinIII  ->  -.  om  ~<_*  A )
 
Theoremisf33lem 8008* Lemma for isfin3-3 8010. (Contributed by Stefan O'Rear, 17-May-2015.)
 |- FinIII  =  { g  |  A. a  e.  ( ~P g  ^m  om ) (
 A. x  e.  om  ( a `  suc  x )  C_  ( a `  x )  ->  |^| ran  a  e.  ran  a ) }
 
Theoremisfin3-2 8009 Weakly Dedekind-infinite sets are exactly those which can be mapped onto  om. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIII  <->  -.  om  ~<_*  A ) )
 
Theoremisfin3-3 8010* Weakly Dedekind-infinite sets are exactly those with an  om-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  ( A  e.  V  ->  ( A  e. FinIII  <->  A. f  e.  ( ~P A  ^m  om )
 ( A. x  e.  om  ( f `  suc  x )  C_  ( f `  x )  ->  |^| ran  f  e.  ran  f ) ) )
 
Theoremfin33i 8011* Inference from isfin3-3 8010. (This is actually a bit stronger than isfin3-3 8010 because it does not assume  F is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e. FinIII  /\  F : om --> ~P A  /\  A. x  e.  om  ( F `
  suc  x )  C_  ( F `  x ) )  ->  |^| ran  F  e.  ran  F )
 
Theoremcompsscnvlem 8012* Lemma for compsscnv 8013. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( x  e. 
 ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) )
 
Theoremcompsscnv 8013* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  `' F  =  F
 
Theoremisf34lem1 8014* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  X  C_  A )  ->  ( F `  X )  =  ( A  \  X ) )
 
Theoremisf34lem2 8015* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( A  e.  V  ->  F : ~P A --> ~P A )
 
Theoremcompssiso 8016* Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
 
Theoremisf34lem3 8017* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
 ( F " X ) )  =  X )
 
Theoremcompss 8018* Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( F " G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
 
Theoremisf34lem4 8019* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  U. X )  =  |^| ( F " X ) )
 
Theoremisf34lem5 8020* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e.  V  /\  ( X  C_  ~P A  /\  X  =/=  (/) ) )  ->  ( F `  |^| X )  = 
 U. ( F " X ) )
 
Theoremisf34lem7 8021* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( ( A  e. FinIII  /\  G : om --> ~P A  /\  A. y  e.  om  ( G `
  y )  C_  ( G `  suc  y
 ) )  ->  U. ran  G  e.  ran  G )
 
Theoremisf34lem6 8022* Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )   =>    |-  ( A  e.  V  ->  ( A  e. FinIII  <->  A. f  e.  ( ~P A  ^m  om )
 ( A. y  e.  om  ( f `  y
 )  C_  ( f `  suc  y )  ->  U. ran  f  e.  ran  f ) ) )
 
Theoremfin34i 8023* Inference from isfin3-4 8024. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e. FinIII  /\  G : om --> ~P A  /\  A. x  e.  om  ( G `
  x )  C_  ( G `  suc  x ) )  ->  U. ran  G  e.  ran  G )
 
Theoremisfin3-4 8024* Weakly Dedekind-infinite sets are exactly those with an  om-indexed ascending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinIII  <->  A. f  e.  ( ~P A  ^m  om )
 ( A. x  e.  om  ( f `  x )  C_  ( f `  suc  x )  ->  U. ran  f  e.  ran  f ) ) )
 
Theoremfin11a 8025 Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  ->  A  e. FinIa )
 
Theoremenfin1ai 8026 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  e. FinIa  ->  B  e. FinIa ) )
 
Theoremisfin1-2 8027 A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  <->  ~P ~P A  e. FinIV )
 
Theoremisfin1-3 8028 A set is I-finite iff every system of subsets contains a maximal subset. Definition I of [Levy58] p. 2. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  `' [ C.] 
 Fr  ~P A ) )
 
Theoremisfin1-4 8029 A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <-> [ C.]  Fr 
 ~P A ) )
 
Theoremdffin1-5 8030 Compact quantifier-free version of the standard definition df-fin 6883. (Contributed by Stefan O'Rear, 6-Jan-2015.)
 |- 
 Fin  =  (  ~~  " om )
 
Theoremfin23 8031 Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness):

Suppose for a contradiction that  A is a set which is II-finite but not III-finite.

For any countable sequence of distinct subsets  T of  A, we can form a decreasing sequence of non-empty subsets  ( U `  T ) by taking finite intersections of initial segments of  T while skipping over any element of  T which would cause the intersection to be empty.

By II-finiteness (as fin2i2 7960) this sequence contains its intersection, call it  Y; since by induction every subset in the sequence  U is non-empty, the intersection must be non-empty.

Suppose that an element  X of  T has non-empty intersection with  Y. Thus, said element has a non-empty intersection with the corresponding element of  U, therefore it was used in the construction of  U and all further elements of  U are subsets of  X, thus  X contains the  Y. That is, all elements of  X either contain  Y or are disjoint from it.

Since there are only two cases, there must exist an infinite subset of  T which uniformly either contain  Y or are disjoint from it. In the former case we can create an infinite set by subtracting  Y from each element. In either case, call the result  Z; this is an infinite set of subsets of 
A, each of which is disjoint from  Y and contained in the union of  T; the union of 
Z is strictly contained in the union of  T, because only the latter is a superset of the non-empty set  Y.

The preceeding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence  B of the  T sets from each stage. Great caution is required to avoid ax-dc 8088 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude  om  e.  _V without the axiom.

This  B sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

 |-  ( A  e. FinII  ->  A  e. FinIII )
 
Theoremfin34 8032 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( A  e. FinIII  ->  A  e. FinIV )
 
Theoremisfin5-2 8033 Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinV  <->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A ) ) ) )
 
Theoremfin45 8034 Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)
 |-  ( A  e. FinIV  ->  A  e. FinV
 )
 
Theoremfin56 8035 Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e. FinV  ->  A  e. FinVI )
 
Theoremfin17 8036 Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  ->  A  e. FinVII )
 
Theoremfin67 8037 Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e. FinVI  ->  A  e. FinVII )
 
Theoremisfin7-2 8038 A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. FinVII  <->  ( A  e.  dom  card  ->  A  e.  Fin ) ) )
 
Theoremfin71num 8039 A well-orderable set is VII-finite iff it is I-finite. Thus, even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  e.  dom  card 
 ->  ( A  e. FinVII  <->  A  e.  Fin ) )
 
Theoremdffin7-2 8040 Class form of isfin7-2 8038. (Contributed by Mario Carneiro, 17-May-2015.)
 |- FinVII  =  ( Fin  u.  ( _V  \  dom  card )
 )
 
Theoremdfacfin7 8041 Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  (CHOICE  <-> FinVII  =  Fin )
 
Theoremfin1a2lem1 8042 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  ( A  e.  On  ->  ( S `  A )  =  suc  A )
 
Theoremfin1a2lem2 8043 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  S : On -1-1-> On
 
Theoremfin1a2lem3 8044 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   =>    |-  ( A  e.  om  ->  ( E `  A )  =  ( 2o  .o  A ) )
 
Theoremfin1a2lem4 8045 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   =>    |-  E : om -1-1-> om
 
Theoremfin1a2lem5 8046 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   =>    |-  ( A  e.  om  ->  ( A  e.  ran  E  <->  -. 
 suc  A  e.  ran  E ) )
 
Theoremfin1a2lem6 8047 Lemma for fin1a2 8057. Establish that  om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   &    |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
 
Theoremfin1a2lem7 8048* Lemma for fin1a2 8057. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  E  =  ( x  e.  om  |->  ( 2o 
 .o  x ) )   &    |-  S  =  ( x  e.  On  |->  suc  x )   =>    |-  (
 ( A  e.  V  /\  A. y  e.  ~P  A ( y  e. FinIII  \/  ( A  \  y
 )  e. FinIII ) )  ->  A  e. FinIII )
 
Theoremfin1a2lem8 8049* Lemma for fin1a2 8057. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
 |-  ( ( A  e.  V  /\  A. x  e. 
 ~P  A ( x  e. FinIII  \/  ( A  \  x )  e. FinIII ) )  ->  A  e. FinIII )
 
Theoremfin1a2lem9 8050* Lemma for fin1a2 8057. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( [ C.]  Or  X  /\  X  C_  Fin  /\  A  e.  om )  ->  { b  e.  X  |  b  ~<_  A }  e.  Fin )
 
Theoremfin1a2lem10 8051 Lemma for fin1a2 8057. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( A  =/=  (/)  /\  A  e.  Fin  /\ [ C.] 
 Or  A )  ->  U. A  e.  A )
 
Theoremfin1a2lem11 8052* Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 8-Nov-2014.)
 |-  ( ( [ C.]  Or  A  /\  A  C_  Fin )  ->  ran  ( b  e. 
 om  |->  U. { c  e.  A  |  c  ~<_  b } )  =  ( A  u.  { (/) } )
 )
 
Theoremfin1a2lem12 8053 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( ( A 
 C_  ~P B  /\ [ C.]  Or  A  /\  -.  U. A  e.  A )  /\  ( A  C_  Fin  /\  A  =/=  (/) ) )  ->  -.  B  e. FinIII )
 
Theoremfin1a2lem13 8054 Lemma for fin1a2 8057. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( ( A 
 C_  ~P B  /\ [ C.]  Or  A  /\  -.  U. A  e.  A )  /\  ( -.  C  e.  Fin  /\  C  e.  A )
 )  ->  -.  ( B  \  C )  e. FinII )
 
Theoremfin12 8055 Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8057. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( A  e.  Fin  ->  A  e. FinII )
 
Theoremfin1a2s 8056* An II-infinite set can have an I-infinite part broken off and remain II-infinite. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e.  V  /\  A. x  e. 
 ~P  A ( x  e.  Fin  \/  ( A  \  x )  e. FinII ) )  ->  A  e. FinII )
 
Theoremfin1a2 8057 Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
 |-  ( A  e. FinIa  ->  A  e. FinII )
 
2.6.13  Hereditarily size-limited sets without Choice
 
Theoremitunifval 8058* Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could concievably be independently useful. (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 )  =  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  A )  |`  om )
 )
 
Theoremitunifn 8059* Functionality of the iterated union. (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 )  Fn  om )
 
Theoremituni0 8060* A zero-fold iterated union. (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 ) `  (/) )  =  A )
 
Theoremitunisuc 8061* Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( ( U `  A ) `  suc  B )  =  U. (
 ( U `  A ) `  B )
 
Theoremitunitc1 8062* Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  U  =  ( x  e.  _V  |->  ( rec ( ( y  e. 
 _V  |->  U. y ) ,  x )  |`  om )
 )   =>    |-  ( ( U `  A ) `  B )  C_  ( TC `  A )
 
Theoremitunitc 8063* The union of all union iterates creates the transitive closure; compare trcl 7426. (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 8064* 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 8065* Lemma for hsmex 8074. 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 8066* Lemma for hsmex 8074. 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 8067* Lemma for hsmex 8074. 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 8068 Lemma for hsmex 8074. 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 8069* Lemma for hsmex 8074. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8213 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 8070* Lemma for hsmex 8074. 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 8071* Lemma for hsmex 8074. 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 8072* Lemma for hsmex 8074. Combining the above constraints, along with itunitc 8063 and tcrank 7570, 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 8073* Lemmr for hsmex 8074. (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 8074* 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 7322. (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 8075* The set of hereditary size-limited sets, assuming ax-reg 7322. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  { s  |  A. x  e.  ( TC ` 
 { s } ) x 
 ~<_  X }  e.  _V )
 
Theoremhsmex3 8076* The set of hereditary size-limited sets, assuming ax-reg 7322, 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 8101, as well as weaker forms such as the axiom of countable choice ax-cc 8077 and dependent choice ax-dc 8088. 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 8077* The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8120, but is weak enough that it can be proven using DC (see axcc 8100). 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 8078* Lemma for axcc2 8079. (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 8079* 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 8080* A possibly more useful version of ax-cc 8077 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 8081* A version of axcc3 8080 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 8082 An ax-cc 8077 equivalent: every set has choice sets of length  om. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |- AC  om  =  _V
 
Theoremaxcc4dom 8083* Relax the constraint on axcc4 8081 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 8084* Lemma for domtriom 8085. (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 8085 Trichotomy of equinumerosity for 
om, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 7956) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( om  ~<_  A  <->  -.  A  ~<  om )
 
Theoremfin41 8086 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 8087 A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8077. See dominfac 8211 for a version proved from ax-ac 8101. The axiom of Regularity is used for this proof, via inf3lem6 7350, 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 8088* 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 8164. 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 8089 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 8090* Lemma for axdc2 8091. 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 8088 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 8091* An apparent strengthening of ax-dc 8088 (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 8092* 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 8093* Lemma for axdc3 8096. 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 8088 (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 8088 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 8094* Simple substitution lemma for axdc3 8096. (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 8095* Lemma for axdc3 8096. 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 8088 (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 8088 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 8091. See axdc3lem2 8093 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 8096* 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 8097* Lemma for axdc4 8098. (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 8098* A more general version of axdc3 8096 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 8099* Lemma for axcc 8100. (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 8100* Although CC can be proven trivially using ac5 8120, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  ( x  ~~  om  ->  E. f A. z  e.  x  ( z  =/= 
 (/)  ->  ( f `  z )  e.  z
 ) )
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