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
Definition	df-djud 7301	The "domain-disjoint-union" of two relations: if R C_ ( A X. X ) and S C_ ( B X. X ) are two binary relations, then ( R ⊔d S ) is the binary relation from ( A ⊔ B ) to X having the universal property of disjoint unions (see updjud 7280 in the case of functions). Remark: the restrictions to dom R (resp. dom S) are not necessary since extra stuff would be thrown away in the post-composition with R (resp. S), as in df-case 7282, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.)
|- ( R ⊔d S ) = ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
Theorem	djufun 7302	The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> Fun F )   &    |- ( ph -> Fun G )   =>    |- ( ph -> Fun ( F ⊔d G ) )
Theorem	djudm 7303	The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.)
|- dom ( F ⊔d G ) = ( dom F ⊔ dom G )
Theorem	djuinj 7304	The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> Fun `' R )   &    |- ( ph -> Fun `' S )   &    |- ( ph -> ( ran R i^i ran S ) = (/) )   =>    |- ( ph -> Fun `' ( R ⊔d S ) )
2.6.36.6  Countable sets
Theorem	0ct 7305	The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- E. f f : om -onto-> ( (/) ⊔ 1o )
Theorem	ctmlemr 7306*	Lemma for ctm 7307. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
|- ( E. x x e. A -> ( E. f f : om -onto-> A -> E. f f : om -onto-> ( A ⊔ 1o ) ) )
Theorem	ctm 7307*	Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( E. x x e. A -> ( E. f f : om -onto-> ( A ⊔ 1o ) <-> E. f f : om -onto-> A ) )
Theorem	ctssdclemn0 7308*	Lemma for ctssdc 7311. The -. (/) e. S case. (Contributed by Jim Kingdon, 16-Aug-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ph -> -. (/) e. S )   =>    |- ( ph -> E. g g : om -onto-> ( A ⊔ 1o ) )
Theorem	ctssdccl 7309*	A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7311 but expressed in terms of classes rather than E.. (Contributed by Jim Kingdon, 30-Oct-2023.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- S = { x e. om | ( F ` x ) e. (inl " A ) }   &    |- G = ( `'inl o. F )   =>    |- ( ph -> ( S C_ om /\ G : S -onto-> A /\ A. n e. om DECID n e. S ) )
Theorem	ctssdclemr 7310*	Lemma for ctssdc 7311. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
|- ( E. f f : om -onto-> ( A ⊔ 1o ) -> E. s ( s C_ om /\ E. f f : s -onto-> A /\ A. n e. om DECID n e. s ) )
Theorem	ctssdc 7311*	A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7348. (Contributed by Jim Kingdon, 15-Aug-2023.)
|- ( E. s ( s C_ om /\ E. f f : s -onto-> A /\ A. n e. om DECID n e. s ) <-> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	enumctlemm 7312*	Lemma for enumct 7313. The case where N is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( ph -> F : N -onto-> A )   &    |- ( ph -> N e. om )   &    |- ( ph -> (/) e. N )   &    |- G = ( k e. om |-> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) )   =>    |- ( ph -> G : om -onto-> A )
Theorem	enumct 7313*	A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as E. n e. om E. f f : n -onto-> A per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as E. g g : om -onto-> ( A ⊔ 1o ) per Definition on page 14:3 of [BauerSwan], p. 3. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( E. n e. om E. f f : n -onto-> A -> E. g g : om -onto-> ( A ⊔ 1o ) )
Theorem	finct 7314*	A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
|- ( A e. Fin -> E. g g : om -onto-> ( A ⊔ 1o ) )
Theorem	omct 7315	om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- E. f f : om -onto-> ( om ⊔ 1o )
Theorem	ctfoex 7316*	A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
|- ( E. f f : om -onto-> ( A ⊔ 1o ) -> A e. _V )
2.6.37  The one-point compactification of the natural numbers This section introduces the one-point compactification of the set of natural numbers, introduced by Escardo as the set of nonincreasing sequences on om with values in 2o. The topological results justifying its name will be proved later.
Syntax	xnninf 7317	Set of nonincreasing sequences in 2o ^m om.
class ℕ∞
Definition	df-nninf 7318*	Define the set of nonincreasing sequences in 2o ^m om. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as NN0* as defined at df-xnn0 9465 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used om or NN0, but the former allows us to take advantage of 2o = { (/) , 1o } (df2o3 6596) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }
Theorem	nninfex 7319	ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
|- ℕ∞ e. _V
Theorem	nninff 7320	An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
|- ( A e. ℕ∞ -> A : om --> 2o )
Theorem	nninfninc 7321	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 7322	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 4773 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 7323	Obsolete version of infnninf 7322 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 7324*	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 7325. (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( N e. om -> ( i e. om |-> if ( i e. N , 1o , (/) ) ) e. ℕ∞ )
Theorem	nnnninf2 7325*	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 7326*	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 7327*	Mapping of a natural number to an element of ℕ∞. Similar to nnnninfeq 7326 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 7328*	Lemma for nninfisol 7331. 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 7329*	Lemma for nninfisol 7331. 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 7330*	Lemma for nninfisol 7331. 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 7331*	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 7378). (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 7332	Extend class definition to include the class of omniscient sets.
class Omni
Definition	df-omni 7333*	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 7334*	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 7335*	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 7336	Lemma for enomni 7337. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
Theorem	enomni 7337	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 6596 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 7338	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 7339	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7340. (Contributed by Jim Kingdon, 29-Jun-2022.)
|- (EXMID -> A. x x e. Omni )
Theorem	exmidomni 7340	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 7341	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
|- (EXMID -> om e. Omni )
Theorem	fodjuomnilemdc 7342*	Lemma for fodjuomni 7347. 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 7343*	Lemma for fodjuomni 7347 and fodjumkv 7358. 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 7344*	Lemma for fodjuomni 7347 and fodjumkv 7358. 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 7345*	Lemma for fodjuomni 7347 and fodjumkv 7358. 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 7346*	Lemma for fodjuomni 7347. 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 7347*	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 7348*	The decidability condition in ctssdc 7311 is needed. More specifically, ctssdc 7311 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 7349	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7350*	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 7351*	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 7352*	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 7353*	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 7354	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 7355	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
|- (EXMID -> om e. Markov )
Theorem	mkvprop 7356*	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 7357*	Lemma for fodjumkv 7358. 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 7358*	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 7359	Lemma for enmkv 7360. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
Theorem	enmkv 7360	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 6596 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 7361	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7362*	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 7363*	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 7364*	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 7365	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 7366	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7365. There is an analogue in terms of analytic omniscience principles at tridceq 16660. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
Theorem	enwomnilem 7367	Lemma for enwomni 7368. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
Theorem	enwomni 7368	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 6596 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 7369*	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 7370*	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 7371*	Lemma for nninfwlpor 7372. 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 7372*	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 7373*	Lemma for nninfwlpoim 7377. (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 7374*	Lemma for nninfwlpoim 7377. (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 7375*	Lemma for nninfwlpoim 7377. (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 7376*	Lemma for nninfinfwlpo 7378. (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 7377*	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 7378*	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 7331). (Contributed by Jim Kingdon, 25-Nov-2025.)
|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )
Theorem	nninfwlpo 7379*	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 7380	Extend class definition to include the cardinal size function.
class card
Syntax	wacn 7381	The axiom of choice for limited-length sequences.
class AC A
Definition	df-card 7382*	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 7383*	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 7384*	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 7385	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 7386	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 7387	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( A e. On -> A e. dom card )
Theorem	cardval3ex 7388*	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 7389*	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 7390	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 7391	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
|- ( card ` (/) ) = (/)
Theorem	ficardon 7392	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 7393*	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 7394	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 7395	Lemma for pr2ne 7396. (Contributed by FL, 17-Aug-2008.)
|- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o )
Theorem	pr2ne 7396	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 7397*	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 7398	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 7399	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 7400	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 )