HomeHome Metamath Proof Explorer
Theorem List (p. 90 of 410)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26627)
  Hilbert Space Explorer  Hilbert Space Explorer
(26628-28150)
  Users' Mathboxes  Users' Mathboxes
(28151-40909)
 

Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremacncc 8901 An ax-cc 8896 equivalent: every set has choice sets of length  om. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |- AC  om  =  _V
 
Theoremaxcc4dom 8902* Relax the constraint on axcc4 8900 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 8903* Lemma for domtriom 8904. (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 8904 Trichotomy of equinumerosity for 
om, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 8775) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
 |-  A  e.  _V   =>    |-  ( om  ~<_  A  <->  -.  A  ~<  om )
 
Theoremfin41 8905 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 8906 A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8896. See dominfac 9029 for a version proved from ax-ac 8920. The axiom of Regularity is used for this proof, via inf3lem6 8169, 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 )
 
3.1.2  Introduce the Axiom of Dependent Choice
 
Axiomax-dc 8907* 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 8982. 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 8908 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 8909* Lemma for axdc2 8910. 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 8907 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 8910* An apparent strengthening of ax-dc 8907 (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 8911* 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 8912* Lemma for axdc3 8915. 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 8907 (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 8907 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 8913* Simple substitution lemma for axdc3 8915. (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 8914* Lemma for axdc3 8915. 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 8907 (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 8907 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 8910. See axdc3lem2 8912 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 8915* 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 8916* Lemma for axdc4 8917. (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 8917* A more general version of axdc3 8915 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 8918* Lemma for axcc 8919. (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 8919* Although CC can be proven trivially using ac5 8938, 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
 ) )
 
3.2  ZFC Set Theory - add the Axiom of Choice
 
3.2.1  Introduce the Axiom of Choice
 
Axiomax-ac 8920* 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 nonempty 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 8923 for a more detailed explanation. Theorem ac2 8922 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 8926 is slightly shorter when the biconditional of ax-ac 8920 is expanded into implication and negation. In axac3 8925 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9137 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8953, ac5 8938, and ac7 8934. The Axiom of Regularity ax-reg 8138 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8592. Equivalents to AC are the well-ordering theorem weth 8956 and Zorn's lemma zorn 8968. See ac4 8936 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8138 for derivation of AC equivalents, we provide ax-ac2 8924 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8924 from ax-ac 8920 is shown by theorem axac2 8927, and the reverse derivation by axac 8928. Therefore, new proofs should normally use ax-ac2 8924 instead. (New usage is discouraged.) (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 8921* Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8920. (New usage is discouraged.) (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 8922* 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 8923 is easier to understand.) Note: aceq0 8580 shows the logical equivalence to ax-ac 8920. (New usage is discouraged.) (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 8923* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8920 can be established by chaining aceq0 8580 and aceq2 8581. A standard textbook version of AC is derived from this one in dfac2a 8591, and this version of AC is derived from the textbook version in dfac2 8592.

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 nonempty 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 8154. The key theorem for this (used in the proof of dfac2 8592) is preleq 8153. 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.

(New usage is discouraged.) (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 8924* In order to avoid uses of ax-reg 8138 for derivation of AC equivalents, we provide ax-ac2 8924, 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 8926. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1680 available. The derivation of ax-ac2 8924 from ax-ac 8920 is shown by theorem axac2 8927, and the reverse derivation by axac 8928. Note that we use ax-reg 8138 to derive ax-ac 8920 from ax-ac2 8924, but not to derive ax-ac2 8924 from ax-ac 8920. (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 8925 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8924 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
 |- CHOICE
 
Theoremackm 8926* A remarkable equivalent to the Axiom of Choice that has only five 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 8627. Maes found this version of AC in April, 2004 (replacing a longer version, also with five 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 8927* Derive ax-ac2 8924 from ax-ac 8920. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (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 8928* Derive ax-ac 8920 from ax-ac2 8924. Note that ax-reg 8138 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 8929 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  (CHOICE  <->  A. x ph )   =>    |-  ph
 
Theoremcardeqv 8930 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 dom  card  =  _V
 
Theoremnumth3 8931 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  dom  card )
 
Theoremnumth2 8932* 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 8933* 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 8934* 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 8935* 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 8936* Equivalent of Axiom of Choice. We do not insist that  f be a function. However, theorem ac5 8938, 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 8952. (Contributed by NM, 21-Jul-1996.)

 |- 
 E. f A. z  e.  x  ( z  =/= 
 (/)  ->  ( f `  z )  e.  z
 )
 
Theoremac4c 8937* 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 8938* 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 8936. (Contributed by NM, 29-Aug-1999.)
 |-  A  e.  _V   =>    |-  E. f ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
 f `  x )  e.  x ) )
 
Theoremac5b 8939* 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 8940* A version of ac6 8941 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 8941* 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 8945, 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 8942* 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 8943* 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 8944* 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 8945* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8395, we derive this strong version of ac6 8941 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 8946* Equivalent of Axiom of Choice. Contrapositive of ac6s 8945. (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 8947* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8948. (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 8948* 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 8949* ac6s 8945 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 8950* Version of ac6 8941 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 8951* 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 8952* 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 8953* 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 8954* 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 8393). (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 8955* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x )
 
Theoremweth 8956* 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 8957* Lemma for zorn2 8967. (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 8958* Lemma for zorn2 8967. (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 8959* Lemma for zorn2 8967. (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 8960* Lemma for zorn2 8967. (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 8961* Lemma for zorn2 8967. (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 8962* Lemma for zorn2 8967. (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 8963* Lemma for zorn2 8967. (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 8964* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8967 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 8965* 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 8968 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 8966* Variant of Zorn's lemma zorng 8965 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 8967* 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 8957 through zorn2lem7 8963; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8963. (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 8968* 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 8967 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 8969* Variant of Zorn's lemma zorn 8968 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 8970* Lemma for ttukey 8979. 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 8971* Lemma for ttukey 8979. 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 8972* Lemma for ttukey 8979. (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 8973* Lemma for ttukey 8979. (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 8974* Lemma for ttukey 8979. 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 8975* Lemma for ttukey 8979. (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 8976* Lemma for ttukey 8979. (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 8977* The Teichmüller-Tukey Lemma ttukey 8979 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 8978* The Teichmüller-Tukey Lemma ttukey 8979 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 8979* 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 8980* Lemma for axdc 8982. (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 ) ) )
 
Theoremaxdclem2 8981* Lemma for axdc 8982. Using the full Axiom of Choice, we can construct a choice function  g on  ~P dom  x. From this, we can build a sequence  F starting at any value  s  e.  dom  x by repeatedly applying  g to the set  ( F `  x ) (where  x is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  F  =  ( rec ( ( y  e. 
 _V  |->  ( g `  { z  |  y x z } )
 ) ,  s )  |`  om )   =>    |-  ( E. z  s x z  ->  ( ran  x  C_  dom  x  ->  E. f A. n  e. 
 om  ( f `  n ) x ( f `  suc  n ) ) )
 
Theoremaxdc 8982* This theorem derives ax-dc 8907 using ax-ac 8920 and ax-inf 8174. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (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 ) )
 
Theoremfodom 8983 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8935. AC is not needed for finite sets - see fodomfi 7881. See also fodomnum 8519. (Contributed by NM, 23-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( F : A -onto-> B  ->  B  ~<_  A )
 
Theoremfodomg 8984 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
 
Theoremfodomb 8985* Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
 |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B 
 ~<_  A ) )
 
Theoremwdomac 8986 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  <->  X  ~<_  Y )
 
Theorembrdom3 8987* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x E* y  x f y  /\  A. x  e.  A  E. y  e.  B  y f x ) )
 
Theorembrdom5 8988* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  x f y  /\  A. x  e.  A  E. y  e.  B  y
 f x ) )
 
Theorembrdom4 8989* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  e.  A  x f y  /\  A. x  e.  A  E. y  e.  B  y f x ) )
 
Theorembrdom7disj 8990* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( A  i^i  B )  =  (/)   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  e.  A  { x ,  y }  e.  f  /\  A. x  e.  A  E. y  e.  B  { y ,  x }  e.  f )
 )
 
Theorembrdom6disj 8991* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( A  i^i  B )  =  (/)   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y { x ,  y }  e.  f  /\  A. x  e.  A  E. y  e.  B  { y ,  x }  e.  f
 ) )
 
Theoremfin71ac 8992 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
 |- FinVII  = 
 Fin
 
Theoremimadomg 8993 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
 
Theoremfnrndomg 8994 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
 |-  ( A  e.  B  ->  ( F  Fn  A  ->  ran  F  ~<_  A ) )
 
Theoremiunfo 8995* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   =>    |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
 
Theoremiundom2g 8996* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. 
B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   &    |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )   &    |-  ( ph  ->  A. x  e.  A  B  ~<_  C )   =>    |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
 
Theoremiundomg 8997* An upper bound for the cardinality of an indexed union, with explicit choice principles.  B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   &    |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )   &    |-  ( ph  ->  A. x  e.  A  B  ~<_  C )   &    |-  ( ph  ->  ( A  X.  C )  e. AC  U_ x  e.  A  B )   =>    |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
 
Theoremiundom 8998* An upper bound for the cardinality of an indexed union.  C depends on  x and should be thought of as  C ( x ). (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B ) 
 ->  U_ x  e.  A  C 
 ~<_  ( A  X.  B ) )
 
Theoremunidom 8999* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  x  ~<_  B ) 
 ->  U. A  ~<_  ( A  X.  B ) )
 
Theoremuniimadom 9000* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-40909
  Copyright terms: Public domain < Previous  Next >