P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nninfninc 7301	All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
|- ( ph -> A e. ℕ∞ )   &    |- ( ph -> X e. om )   &    |- ( ph -> Y e. om )   &    |- ( ph -> X C_ Y )   &    |- ( ph -> ( A ` X ) = (/) )   =>    |- ( ph -> ( A ` Y ) = (/) )
Theorem	infnninf 7302	The point at infinity in ℕ∞ is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as ( om X. { 1o } ), as fconstmpt 4766 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
|- ( i e. om |-> 1o ) e. ℕ∞
Theorem	infnninfOLD 7303	Obsolete version of infnninf 7302 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( om X. { 1o } ) e. ℕ∞
Theorem	nnnninf 7304*	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 7305. (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( N e. om -> ( i e. om |-> if ( i e. N , 1o , (/) ) ) e. ℕ∞ )
Theorem	nnnninf2 7305*	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. ℕ∞ )
Theorem	nnnninfeq 7306*	Mapping of a natural number to an element of ℕ∞. (Contributed by Jim Kingdon, 4-Aug-2022.)
|- ( ph -> P e. ℕ∞ )   &    |- ( 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 , (/) ) ) )
Theorem	nnnninfeq2 7307*	Mapping of a natural number to an element of ℕ∞. Similar to nnnninfeq 7306 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. ℕ∞ )   &    |- ( ph -> N e. om )   &    |- ( ph -> ( P ` U. N ) = 1o )   &    |- ( ph -> ( P ` N ) = (/) )   =>    |- ( ph -> P = ( i e. om |-> if ( i e. N , 1o , (/) ) ) )
Theorem	nninfisollem0 7308*	Lemma for nninfisol 7311. The case where N is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
|- ( ph -> X e. ℕ∞ )   &    |- ( ph -> ( X ` N ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> N = (/) )   =>    |- ( ph -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
Theorem	nninfisollemne 7309*	Lemma for nninfisol 7311. A case where N is a successor and N and X are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
|- ( ph -> X e. ℕ∞ )   &    |- ( ph -> ( X ` N ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> N =/= (/) )   &    |- ( ph -> ( X ` U. N ) = (/) )   =>    |- ( ph -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
Theorem	nninfisollemeq 7310*	Lemma for nninfisol 7311. The case where N is a successor and N and X are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
|- ( ph -> X e. ℕ∞ )   &    |- ( 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 )
Theorem	nninfisol 7311*	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). By contrast, the point at infinity being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO) (nninfinfwlpo 7358). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
|- ( ( N e. om /\ X e. ℕ∞ ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
2.6.38  Omniscient sets
Syntax	comni 7312	Extend class definition to include the class of omniscient sets.
class Omni
Definition	df-omni 7313*	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 ) ) }
Theorem	isomni 7314*	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 ) ) ) )
Theorem	isomnimap 7315*	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 ) ) )
Theorem	enomnilem 7316	Lemma for enomni 7317. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
Theorem	enomni 7317	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 6583 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 ) )
Theorem	finomni 7318	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 )
Theorem	exmidomniim 7319	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7320. (Contributed by Jim Kingdon, 29-Jun-2022.)
|- (EXMID -> A. x x e. Omni )
Theorem	exmidomni 7320	Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
|- (EXMID <-> A. x x e. Omni )
Theorem	exmidlpo 7321	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
|- (EXMID -> om e. Omni )
Theorem	fodjuomnilemdc 7322*	Lemma for fodjuomni 7327. 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 ) )
Theorem	fodjuf 7323*	Lemma for fodjuomni 7327 and fodjumkv 7338. 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 ) )
Theorem	fodjum 7324*	Lemma for fodjuomni 7327 and fodjumkv 7338. 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 )
Theorem	fodju0 7325*	Lemma for fodjuomni 7327 and fodjumkv 7338. 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 = (/) )
Theorem	fodjuomnilemres 7326*	Lemma for fodjuomni 7327. 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 = (/) ) )
Theorem	fodjuomni 7327*	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 = (/) ) )
Theorem	ctssexmid 7328*	The decidability condition in ctssdc 7291 is needed. More specifically, ctssdc 7291 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
Syntax	cmarkov 7329	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7330*	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 ) = (/) ) ) }
Theorem	ismkv 7331*	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 ) = (/) ) ) ) )
Theorem	ismkvmap 7332*	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 ) = (/) ) ) )
Theorem	ismkvnex 7333*	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 ) ) )
Theorem	omnimkv 7334	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 )
Theorem	exmidmp 7335	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
|- (EXMID -> om e. Markov )
Theorem	mkvprop 7336*	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 )
Theorem	fodjumkvlemres 7337*	Lemma for fodjumkv 7338. 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 ) )
Theorem	fodjumkv 7338*	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 ) )
Theorem	enmkvlem 7339	Lemma for enmkv 7340. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
Theorem	enmkv 7340	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 6583 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
Syntax	cwomni 7341	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7342*	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 ) }
Theorem	iswomni 7343*	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 ) ) )
Theorem	iswomnimap 7344*	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 ) )
Theorem	omniwomnimkv 7345	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 ) )
Theorem	lpowlpo 7346	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7345. There is an analogue in terms of analytic omniscience principles at tridceq 16484. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
Theorem	enwomnilem 7347	Lemma for enwomni 7348. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
Theorem	enwomni 7348	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 6583 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 ) )
Theorem	nninfdcinf 7349*	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. ℕ∞ )   =>    |- ( ph -> DECID N = ( i e. om |-> 1o ) )
Theorem	nninfwlporlemd 7350*	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 ) ) )
Theorem	nninfwlporlem 7351*	Lemma for nninfwlpor 7352. 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 )
Theorem	nninfwlpor 7352*	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. ℕ∞ A. y e. ℕ∞ DECID x = y )
Theorem	nninfwlpoimlemg 7353*	Lemma for nninfwlpoim 7357. (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. ℕ∞ )
Theorem	nninfwlpoimlemginf 7354*	Lemma for nninfwlpoim 7357. (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 ) )
Theorem	nninfwlpoimlemdc 7355*	Lemma for nninfwlpoim 7357. (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. ℕ∞ A. y e. ℕ∞ DECID x = y )   =>    |- ( ph -> DECID A. n e. om ( F ` n ) = 1o )
Theorem	nninfinfwlpolem 7356*	Lemma for nninfinfwlpo 7358. (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. ℕ∞ DECID x = ( i e. om |-> 1o ) )   =>    |- ( ph -> DECID A. n e. om ( F ` n ) = 1o )
Theorem	nninfwlpoim 7357*	Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
|- ( A. x e. ℕ∞ A. y e. ℕ∞ DECID x = y -> om e. WOmni )
Theorem	nninfinfwlpo 7358*	The point at infinity in ℕ∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ∞ corresponding to natural numbers are isolated (nninfisol 7311). (Contributed by Jim Kingdon, 25-Nov-2025.)
|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )
Theorem	nninfwlpo 7359*	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. ℕ∞ A. y e. ℕ∞ DECID x = y <-> om e. WOmni )
2.6.41  Cardinal numbers
Syntax	ccrd 7360	Extend class definition to include the cardinal size function.
class card
Syntax	wacn 7361	The axiom of choice for limited-length sequences.
class AC A
Definition	df-card 7362*	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 } )
Definition	df-acnm 7363*	Define a local and length-limited version of the axiom of choice. The definition of the predicate X e. AC A is that for all families of inhabited subsets of X indexed on A (i.e. functions A --> { z e. ~P X | E. j j e. z }), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) Change nonempty to inhabited. (Revised by Jim Kingdon, 22-Nov-2025.)
|- AC A = { x | ( A e. _V /\ A. f e. ( { z e. ~P x | E. j j e. z } ^m A ) E. g A. y e. A ( g ` y ) e. ( f ` y ) ) }
Theorem	cardcl 7364*	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 )
Theorem	isnumi 7365	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. On /\ A ~~ B ) -> B e. dom card )
Theorem	finnum 7366	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 )
Theorem	onenon 7367	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( A e. On -> A e. dom card )
Theorem	cardval3ex 7368*	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 } )
Theorem	oncardval 7369*	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 } )
Theorem	cardonle 7370	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 )
Theorem	card0 7371	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
|- ( card ` (/) ) = (/)
Theorem	ficardon 7372	The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
|- ( A e. Fin -> ( card ` A ) e. On )
Theorem	carden2bex 7373*	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 ) )
Theorem	pm54.43 7374	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 ) )
Theorem	pr2nelem 7375	Lemma for pr2ne 7376. (Contributed by FL, 17-Aug-2008.)
|- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o )
Theorem	pr2ne 7376	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 ) )
Theorem	en2prde 7377*	A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by Jim Kingdon, 11-Jan-2026.)
|- ( V ~~ 2o -> E. a E. b ( a =/= b /\ V = { a , b } ) )
Theorem	pr1or2 7378	An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
|- ( ( A e. C /\ B e. D /\ DECID A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
Theorem	pr2cv1 7379	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
|- ( { A , B } ~~ 2o -> A e. _V )
Theorem	pr2cv2 7380	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
|- ( { A , B } ~~ 2o -> B e. _V )
Theorem	pr2cv 7381	If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
|- ( { A , B } ~~ 2o -> ( A e. _V /\ B e. _V ) )
Theorem	exmidonfinlem 7382*	Lemma for exmidonfin 7383. (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 )
Theorem	exmidonfin 7383	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 7042 and nnon 4702. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
|- ( om = ( On i^i Fin ) -> EXMID )
Theorem	en2eleq 7384	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 } ) } )
Theorem	en2other2 7385	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 )
Theorem	dju1p1e2 7386	Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
|- ( 1o ⊔ 1o ) ~~ 2o
Theorem	infpwfidom 7387	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 ) )
Theorem	exmidfodomrlemeldju 7388	Lemma for exmidfodomr 7393. A variant of djur 7247. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( ph -> A C_ 1o )   &    |- ( ph -> B e. ( A ⊔ 1o ) )   =>    |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
Theorem	exmidfodomrlemreseldju 7389	Lemma for exmidfodomrlemrALT 7392. A variant of eldju 7246. (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 ) ` (/) ) ) )
Theorem	exmidfodomrlemim 7390*	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 ) )
Theorem	exmidfodomrlemr 7391*	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 )
Theorem	exmidfodomrlemrALT 7392*	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 7391. In particular, this proof uses eldju 7246 instead of djur 7247 and avoids djulclb 7233. (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 )
Theorem	exmidfodomr 7393*	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 ) )
Theorem	acnrcl 7394	Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( X e. AC A -> A e. _V )
Theorem	acneq 7395	Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( A = C -> AC A = AC C )
Theorem	isacnm 7396*	The property of being a choice set of length A. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( ( X e. V /\ A e. W ) -> ( X e. AC A <-> A. f e. ( { z e. ~P X | E. j j e. z } ^m A ) E. g A. x e. A ( g ` x ) e. ( f ` x ) ) )
Theorem	finacn 7397	Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( A e. Fin -> AC A = _V )
2.6.42  Axiom of Choice equivalents
Syntax	wac 7398	Formula for an abbreviation of the axiom of choice.
wff CHOICE
Definition	df-ac 7399*	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 4629 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 ) )
Theorem	acfun 7400*	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 ) )