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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoreminfnninfOLD 7101 Obsolete version of infnninf 7100 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  X.  { 1o } )  e.
 
Theoremnnnninf 7102* Elements of ℕ corresponding to natural numbers. The natural number  N corresponds to a sequence of  N ones followed by zeroes. This can be strengthened to include infinity, see nnnninf2 7103. (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  ( N  e.  om  ->  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  e. )
 
Theoremnnnninf2 7103* Canonical embedding of  suc  om into ℕ. (Contributed by BJ, 10-Aug-2024.)
 |-  ( N  e.  suc  om 
 ->  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  e. )
 
Theoremnnnninfeq 7104* Mapping of a natural number to an element of ℕ. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( ph  ->  P  e. )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  A. x  e.  N  ( P `  x )  =  1o )   &    |-  ( ph  ->  ( P `  N )  =  (/) )   =>    |-  ( ph  ->  P  =  ( i  e. 
 om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )
 
Theoremnnnninfeq2 7105* Mapping of a natural number to an element of ℕ. Similar to nnnninfeq 7104 but if we have information about a single  1o digit, that gives information about all previous digits. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( ph  ->  P  e. )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  ( P `  U. N )  =  1o )   &    |-  ( ph  ->  ( P `  N )  =  (/) )   =>    |-  ( ph  ->  P  =  ( i  e. 
 om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )
 
Theoremnninfisollem0 7106* Lemma for nninfisol 7109. The case where  N is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
 |-  ( ph  ->  X  e. )   &    |-  ( ph  ->  ( X `  N )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  N  =  (/) )   =>    |-  ( ph  -> DECID  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  =  X )
 
Theoremnninfisollemne 7107* Lemma for nninfisol 7109. A case where  N is a successor and  N and  X are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
 |-  ( ph  ->  X  e. )   &    |-  ( ph  ->  ( X `  N )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  N  =/=  (/) )   &    |-  ( ph  ->  ( X `  U. N )  =  (/) )   =>    |-  ( ph  -> DECID  ( i  e.  om  |->  if (
 i  e.  N ,  1o ,  (/) ) )  =  X )
 
Theoremnninfisollemeq 7108* Lemma for nninfisol 7109. The case where  N is a successor and  N and  X are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
 |-  ( ph  ->  X  e. )   &    |-  ( ph  ->  ( X `  N )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  N  =/=  (/) )   &    |-  ( ph  ->  ( X `  U. N )  =  1o )   =>    |-  ( ph  -> DECID 
 ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  =  X )
 
Theoremnninfisol 7109* Finite elements of ℕ are isolated. That is, given a natural number and any element of ℕ, it is decidable whether the natural number (when converted to an element of ℕ) is equal to the given element of ℕ. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence  X to decide whether it is equal to  N (in fact, you only need to look at two elements and  N tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
 |-  ( ( N  e.  om 
 /\  X  e. )  -> DECID  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  =  X )
 
2.6.38  Omniscient sets
 
Syntaxcomni 7110 Extend class definition to include the class of omniscient sets.
 class Omni
 
Definitiondf-omni 7111* An omniscient set is one where we can decide whether a predicate (here represented by a function  f) holds (is equal to  1o) for all elements or fails to hold (is equal to  (/)) for some element. Definition 3.1 of [Pierik], p. 14.

In particular,  om  e. Omni is known as the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 28-Jun-2022.)

 |- Omni  =  { y  |  A. f ( f : y --> 2o  ->  ( E. x  e.  y  (
 f `  x )  =  (/)  \/  A. x  e.  y  ( f `  x )  =  1o ) ) }
 
Theoremisomni 7112* The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
 |-  ( A  e.  V  ->  ( A  e. Omni  <->  A. f ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x )  =  (/)  \/  A. x  e.  A  (
 f `  x )  =  1o ) ) ) )
 
Theoremisomnimap 7113* The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.)
 |-  ( A  e.  V  ->  ( A  e. Omni  <->  A. f  e.  ( 2o  ^m  A ) ( E. x  e.  A  ( f `  x )  =  (/)  \/  A. x  e.  A  (
 f `  x )  =  1o ) ) )
 
Theoremenomnilem 7114 Lemma for enomni 7115. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
 |-  ( A  ~~  B  ->  ( A  e. Omni  ->  B  e. Omni ) )
 
Theoremenomni 7115 Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either  om  e. Omni or  NN0  e. Omni. The former is a better match to conventional notation in the sense that df2o3 6409 says that  2o  =  { (/)
,  1o } whereas the corresponding relationship does not exist between  2 and  { 0 ,  1 }. (Contributed by Jim Kingdon, 13-Jul-2022.)
 |-  ( A  ~~  B  ->  ( A  e. Omni  <->  B  e. Omni ) )
 
Theoremfinomni 7116 A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)
 |-  ( A  e.  Fin  ->  A  e. Omni )
 
Theoremexmidomniim 7117 Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7118. (Contributed by Jim Kingdon, 29-Jun-2022.)
 |-  (EXMID 
 ->  A. x  x  e. Omni
 )
 
Theoremexmidomni 7118 Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
 |-  (EXMID  <->  A. x  x  e. Omni )
 
Theoremexmidlpo 7119 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
 |-  (EXMID 
 ->  om  e. Omni )
 
Theoremfodjuomnilemdc 7120* Lemma for fodjuomni 7125. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
 |-  ( ph  ->  F : O -onto-> ( A B ) )   =>    |-  ( ( ph  /\  X  e.  O )  -> DECID  E. z  e.  A  ( F `  X )  =  (inl `  z
 ) )
 
Theoremfodjuf 7121* Lemma for fodjuomni 7125 and fodjumkv 7136. Domain and range of  P. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  F : O -onto-> ( A B ) )   &    |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   &    |-  ( ph  ->  O  e.  V )   =>    |-  ( ph  ->  P  e.  ( 2o  ^m  O ) )
 
Theoremfodjum 7122* Lemma for fodjuomni 7125 and fodjumkv 7136. A condition which shows that  A is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  F : O -onto-> ( A B ) )   &    |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   &    |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )   =>    |-  ( ph  ->  E. x  x  e.  A )
 
Theoremfodju0 7123* Lemma for fodjuomni 7125 and fodjumkv 7136. A condition which shows that  A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  F : O -onto-> ( A B ) )   &    |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   &    |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )   =>    |-  ( ph  ->  A  =  (/) )
 
Theoremfodjuomnilemres 7124* Lemma for fodjuomni 7125. The final result with  P expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)
 |-  ( ph  ->  O  e. Omni )   &    |-  ( ph  ->  F : O -onto-> ( A B ) )   &    |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   =>    |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
 
Theoremfodjuomni 7125* A condition which ensures  A is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
 |-  ( ph  ->  O  e. Omni )   &    |-  ( ph  ->  F : O -onto-> ( A B ) )   =>    |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
 
Theoremctssexmid 7126* The decidability condition in ctssdc 7090 is needed. More specifically, ctssdc 7090 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
 |-  ( ( y  C_  om 
 /\  E. f  f : y -onto-> x )  ->  E. f  f : om -onto-> ( x 1o ) )   &    |-  om  e. Omni   =>    |-  ( ph  \/  -.  ph )
 
2.6.39  Markov's principle
 
Syntaxcmarkov 7127 Extend class definition to include the class of Markov sets.
 class Markov
 
Definitiondf-markov 7128* A Markov set is one where if a predicate (here represented by a function  f) on that set does not hold (where hold means is equal to  1o) for all elements, then there exists an element where it fails (is equal to  (/)). Generalization of definition 2.5 of [Pierik], p. 9.

In particular,  om  e. Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)

 |- Markov  =  { y  |  A. f ( f : y --> 2o  ->  ( -. 
 A. x  e.  y  ( f `  x )  =  1o  ->  E. x  e.  y  ( f `  x )  =  (/) ) ) }
 
Theoremismkv 7129* The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
 |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f ( f : A --> 2o  ->  ( -.  A. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
 
Theoremismkvmap 7130* The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
 |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f  e.  ( 2o  ^m  A ) ( -.  A. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) )
 
Theoremismkvnex 7131* The predicate of being Markov stated in terms of double negation and comparison with  1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
 |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f  e.  ( 2o  ^m  A ) ( -.  -.  E. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  1o )
 ) )
 
Theoremomnimkv 7132 An omniscient set is Markov. In particular, the case where  A is  om means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
 |-  ( A  e. Omni  ->  A  e. Markov )
 
Theoremexmidmp 7133 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
 |-  (EXMID 
 ->  om  e. Markov )
 
Theoremmkvprop 7134* Markov's Principle expressed in terms of propositions (or more precisely, the  A  =  om case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
 |-  ( ( A  e. Markov  /\ 
 A. n  e.  A DECID  ph  /\  -.  A. n  e.  A  -.  ph )  ->  E. n  e.  A  ph )
 
Theoremfodjumkvlemres 7135* Lemma for fodjumkv 7136. The final result with  P expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  M  e. Markov )   &    |-  ( ph  ->  F : M -onto-> ( A B ) )   &    |-  P  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   =>    |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
 )
 
Theoremfodjumkv 7136* A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  M  e. Markov )   &    |-  ( ph  ->  F : M -onto-> ( A B ) )   =>    |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
 )
 
Theoremenmkvlem 7137 Lemma for enmkv 7138. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. Markov  ->  B  e. Markov ) )
 
Theoremenmkv 7138 Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either  om  e. Markov or  NN0  e. Markov. The former is a better match to conventional notation in the sense that df2o3 6409 says that  2o  =  { (/)
,  1o } whereas the corresponding relationship does not exist between  2 and  { 0 ,  1 }. (Contributed by Jim Kingdon, 24-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. Markov  <->  B  e. Markov ) )
 
2.6.40  Weakly omniscient sets
 
Syntaxcwomni 7139 Extend class definition to include the class of weakly omniscient sets.
 class WOmni
 
Definitiondf-womni 7140* A weakly omniscient set is one where we can decide whether a predicate (here represented by a function  f) holds (is equal to  1o) for all elements or not. Generalization of definition 2.4 of [Pierik], p. 9.

In particular,  om  e. WOmni is known as the Weak Limited Principle of Omniscience (WLPO).

The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.)

 |- WOmni  =  { y  |  A. f ( f : y --> 2o  -> DECID  A. x  e.  y  ( f `  x )  =  1o ) }
 
Theoremiswomni 7141* The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
 |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x )  =  1o ) ) )
 
Theoremiswomnimap 7142* The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
 |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f  e.  ( 2o  ^m  A )DECID  A. x  e.  A  ( f `  x )  =  1o ) )
 
Theoremomniwomnimkv 7143 A set is omniscient if and only if it is weakly omniscient and Markov. The case  A  =  om says that LPO  <-> WLPO  /\ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
 |-  ( A  e. Omni  <->  ( A  e. WOmni  /\  A  e. Markov ) )
 
Theoremlpowlpo 7144 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7143. There is an analogue in terms of analytic omniscience principles at tridceq 14088. (Contributed by Jim Kingdon, 24-Jul-2024.)
 |-  ( om  e. Omni  ->  om  e. WOmni )
 
Theoremenwomnilem 7145 Lemma for enwomni 7146. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. WOmni  ->  B  e. WOmni ) )
 
Theoremenwomni 7146 Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either  om  e. WOmni or  NN0  e. WOmni. The former is a better match to conventional notation in the sense that df2o3 6409 says that  2o  =  { (/)
,  1o } whereas the corresponding relationship does not exist between  2 and  { 0 ,  1 }. (Contributed by Jim Kingdon, 20-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. WOmni  <->  B  e. WOmni ) )
 
Theoremnninfdcinf 7147* The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
 |-  ( ph  ->  om  e. WOmni )   &    |-  ( ph  ->  N  e. )   =>    |-  ( ph  -> DECID  N  =  ( i  e.  om  |->  1o ) )
 
Theoremnninfwlporlemd 7148* Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)
 |-  ( ph  ->  X : om --> 2o )   &    |-  ( ph  ->  Y : om --> 2o )   &    |-  D  =  ( i  e.  om  |->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )   =>    |-  ( ph  ->  ( X  =  Y  <->  D  =  (
 i  e.  om  |->  1o ) ) )
 
Theoremnninfwlporlem 7149* Lemma for nninfwlpor 7150. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
 |-  ( ph  ->  X : om --> 2o )   &    |-  ( ph  ->  Y : om --> 2o )   &    |-  D  =  ( i  e.  om  |->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )   &    |-  ( ph  ->  om  e. WOmni )   =>    |-  ( ph  -> DECID  X  =  Y )
 
Theoremnninfwlpor 7150* The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
 |-  ( om  e. WOmni  ->  A. x  e.  A. y  e. DECID  x  =  y )
 
Theoremnninfwlpoimlemg 7151* Lemma for nninfwlpoim 7154. (Contributed by Jim Kingdon, 8-Dec-2024.)
 |-  ( ph  ->  F : om --> 2o )   &    |-  G  =  ( i  e.  om  |->  if ( E. x  e. 
 suc  i ( F `
  x )  =  (/) ,  (/) ,  1o )
 )   =>    |-  ( ph  ->  G  e. )
 
Theoremnninfwlpoimlemginf 7152* Lemma for nninfwlpoim 7154. (Contributed by Jim Kingdon, 8-Dec-2024.)
 |-  ( ph  ->  F : om --> 2o )   &    |-  G  =  ( i  e.  om  |->  if ( E. x  e. 
 suc  i ( F `
  x )  =  (/) ,  (/) ,  1o )
 )   =>    |-  ( ph  ->  ( G  =  ( i  e.  om  |->  1o )  <->  A. n  e.  om  ( F `  n )  =  1o ) )
 
Theoremnninfwlpoimlemdc 7153* Lemma for nninfwlpoim 7154. (Contributed by Jim Kingdon, 8-Dec-2024.)
 |-  ( ph  ->  F : om --> 2o )   &    |-  G  =  ( i  e.  om  |->  if ( E. x  e. 
 suc  i ( F `
  x )  =  (/) ,  (/) ,  1o )
 )   &    |-  ( ph  ->  A. x  e.  A. y  e. DECID  x  =  y )   =>    |-  ( ph  -> DECID  A. n  e.  om  ( F `  n )  =  1o )
 
Theoremnninfwlpoim 7154* Decidable equality for ℕ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
 |-  ( A. x  e.  A. y  e. DECID  x  =  y  ->  om  e. WOmni )
 
Theoremnninfwlpo 7155* Decidability of equality for ℕ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
 |-  ( A. x  e.  A. y  e. DECID  x  =  y  <->  om  e. WOmni )
 
2.6.41  Cardinal numbers
 
Syntaxccrd 7156 Extend class definition to include the cardinal size function.
 class  card
 
Definitiondf-card 7157* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
 |- 
 card  =  ( x  e.  _V  |->  |^| { y  e. 
 On  |  y  ~~  x } )
 
Theoremcardcl 7158* The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( E. y  e. 
 On  y  ~~  A  ->  ( card `  A )  e.  On )
 
Theoremisnumi 7159 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card
 )
 
Theoremfinnum 7160 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  Fin  ->  A  e.  dom  card )
 
Theoremonenon 7161 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  On  ->  A  e.  dom  card )
 
Theoremcardval3ex 7162* The value of  ( card `  A
). (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( E. x  e. 
 On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e. 
 On  |  y  ~~  A } )
 
Theoremoncardval 7163* The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( A  e.  On  ->  ( card `  A )  =  |^| { x  e. 
 On  |  x  ~~  A } )
 
Theoremcardonle 7164 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  On  ->  ( card `  A )  C_  A )
 
Theoremcard0 7165 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
 |-  ( card `  (/) )  =  (/)
 
Theoremcarden2bex 7166* If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( ( A  ~~  B  /\  E. x  e. 
 On  x  ~~  A )  ->  ( card `  A )  =  ( card `  B ) )
 
Theorempm54.43 7167 Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
 |-  ( ( A  ~~  1o  /\  B  ~~  1o )  ->  ( ( A  i^i  B )  =  (/) 
 <->  ( A  u.  B )  ~~  2o ) )
 
Theorempr2nelem 7168 Lemma for pr2ne 7169. (Contributed by FL, 17-Aug-2008.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B ) 
 ->  { A ,  B }  ~~  2o )
 
Theorempr2ne 7169 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )
 
Theoremexmidonfinlem 7170* Lemma for exmidonfin 7171. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
 |-  A  =  { { x  e.  { (/) }  |  ph
 } ,  { x  e.  { (/) }  |  -.  ph
 } }   =>    |-  ( om  =  ( On  i^i  Fin )  -> DECID  ph )
 
Theoremexmidonfin 7171 If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6850 and nnon 4594. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
 |-  ( om  =  ( On  i^i  Fin )  -> EXMID )
 
Theoremen2eleq 7172 Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P 
 \  { X }
 ) } )
 
Theoremen2other2 7173 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { U. ( P  \  { X } ) }
 )  =  X )
 
Theoremdju1p1e2 7174 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( 1o 1o )  ~~  2o
 
Theoreminfpwfidom 7175 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
 
Theoremexmidfodomrlemeldju 7176 Lemma for exmidfodomr 7181. A variant of djur 7046. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
 
Theoremexmidfodomrlemreseldju 7177 Lemma for exmidfodomrlemrALT 7180. A variant of eldju 7045. (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  (
 ( (/)  e.  A  /\  B  =  ( (inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `  (/) ) ) )
 
Theoremexmidfodomrlemim 7178* Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID 
 ->  A. x A. y
 ( ( E. z  z  e.  y  /\  y 
 ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
Theoremexmidfodomrlemr 7179* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
 
TheoremexmidfodomrlemrALT 7180* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7179. In particular, this proof uses eldju 7045 instead of djur 7046 and avoids djulclb 7032. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
 
Theoremexmidfodomr 7181* Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID  <->  A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
2.6.42  Axiom of Choice equivalents
 
Syntaxwac 7182 Formula for an abbreviation of the axiom of choice.
 wff CHOICE
 
Definitiondf-ac 7183* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There are some decisions about how to write this definition especially around whether ax-setind 4521 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.)

 |-  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
 )
 
Theoremacfun 7184* A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
 |-  ( ph  -> CHOICE )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x  e.  A  E. w  w  e.  x )   =>    |-  ( ph  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  x )
 )
 
Theoremexmidaclem 7185* Lemma for exmidac 7186. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  y  =  { (/) } ) }   &    |-  B  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  y  =  { (/) } ) }   &    |-  C  =  { A ,  B }   =>    |-  (CHOICE 
 -> EXMID )
 
Theoremexmidac 7186 The axiom of choice implies excluded middle. See acexmid 5852 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
 |-  (CHOICE 
 -> EXMID )
 
2.6.43  Cardinal number arithmetic
 
Theoremendjudisj 7187 Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( A B )  ~~  ( A  u.  B ) )
 
Theoremdjuen 7188 Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A C ) 
 ~~  ( B D ) )
 
Theoremdjuenun 7189 Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
 |-  ( ( A  ~~  B  /\  C  ~~  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A C )  ~~  ( B  u.  D ) )
 
Theoremdju1en 7190 Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  -.  A  e.  A )  ->  ( A 1o )  ~~  suc  A )
 
Theoremdju0en 7191 Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A (/) )  ~~  A )
 
Theoremxp2dju 7192 Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( 2o  X.  A )  =  ( A A )
 
Theoremdjucomen 7193 Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B ) 
 ~~  ( B A ) )
 
Theoremdjuassen 7194 Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( ( A B ) C )  ~~  ( A ( B C ) ) )
 
Theoremxpdjuen 7195 Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( A  X.  ( B C ) )  ~~  ( ( A  X.  B ) ( A  X.  C ) ) )
 
Theoremdjudoml 7196 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  ( A B ) )
 
Theoremdjudomr 7197 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  ~<_  ( A B ) )
 
2.6.44  Ordinal trichotomy
 
Theoremexmidontriimlem1 7198 Lemma for exmidontriim 7202. A variation of r19.30dc 2617. (Contributed by Jim Kingdon, 12-Aug-2024.)
 |-  ( ( A. x  e.  A  ( ph  \/  ps 
 \/  ch )  /\ EXMID )  ->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps  \/  A. x  e.  A  ch ) )
 
Theoremexmidontriimlem2 7199* Lemma for exmidontriim 7202. (Contributed by Jim Kingdon, 12-Aug-2024.)
 |-  ( ph  ->  B  e.  On )   &    |-  ( ph  -> EXMID )   &    |-  ( ph  ->  A. y  e.  B  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) )   =>    |-  ( ph  ->  ( A  e.  B  \/  A. y  e.  B  y  e.  A ) )
 
Theoremexmidontriimlem3 7200* Lemma for exmidontriim 7202. What we get to do based on induction on both  A and  B. (Contributed by Jim Kingdon, 10-Aug-2024.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  -> EXMID
 )   &    |-  ( ph  ->  A. z  e.  A  A. y  e. 
 On  ( z  e.  y  \/  z  =  y  \/  y  e.  z ) )   &    |-  ( ph  ->  A. y  e.  B  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) )   =>    |-  ( ph  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
 )
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