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	enomnilem 7301	Lemma for enomni 7302. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
Theorem	enomni 7302	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 6574 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 7303	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 7304	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7305. (Contributed by Jim Kingdon, 29-Jun-2022.)
|- (EXMID -> A. x x e. Omni )
Theorem	exmidomni 7305	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 7306	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
|- (EXMID -> om e. Omni )
Theorem	fodjuomnilemdc 7307*	Lemma for fodjuomni 7312. 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 7308*	Lemma for fodjuomni 7312 and fodjumkv 7323. 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 7309*	Lemma for fodjuomni 7312 and fodjumkv 7323. 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 7310*	Lemma for fodjuomni 7312 and fodjumkv 7323. 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 7311*	Lemma for fodjuomni 7312. 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 7312*	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 7313*	The decidability condition in ctssdc 7276 is needed. More specifically, ctssdc 7276 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 7314	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7315*	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 7316*	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 7317*	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 7318*	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 7319	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 7320	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
|- (EXMID -> om e. Markov )
Theorem	mkvprop 7321*	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 7322*	Lemma for fodjumkv 7323. 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 7323*	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 7324	Lemma for enmkv 7325. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
Theorem	enmkv 7325	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 6574 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 7326	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7327*	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 7328*	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 7329*	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 7330	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 7331	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7330. There is an analogue in terms of analytic omniscience principles at tridceq 16383. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
Theorem	enwomnilem 7332	Lemma for enwomni 7333. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
Theorem	enwomni 7333	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 6574 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 7334*	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 7335*	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 7336*	Lemma for nninfwlpor 7337. 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 7337*	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 7338*	Lemma for nninfwlpoim 7342. (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 7339*	Lemma for nninfwlpoim 7342. (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 7340*	Lemma for nninfwlpoim 7342. (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 7341*	Lemma for nninfinfwlpo 7343. (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 7342*	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 7343*	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 7296). (Contributed by Jim Kingdon, 25-Nov-2025.)
|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )
Theorem	nninfwlpo 7344*	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 7345	Extend class definition to include the cardinal size function.
class card
Syntax	wacn 7346	The axiom of choice for limited-length sequences.
class AC A
Definition	df-card 7347*	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 7348*	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 7349*	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 7350	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 7351	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 7352	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( A e. On -> A e. dom card )
Theorem	cardval3ex 7353*	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 7354*	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 7355	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 7356	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
|- ( card ` (/) ) = (/)
Theorem	ficardon 7357	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 7358*	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 7359	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 7360	Lemma for pr2ne 7361. (Contributed by FL, 17-Aug-2008.)
|- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o )
Theorem	pr2ne 7361	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 7362*	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 7363	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 7364	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 7365	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 7366	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 7367*	Lemma for exmidonfin 7368. (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 7368	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 7030 and nnon 4701. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
|- ( om = ( On i^i Fin ) -> EXMID )
Theorem	en2eleq 7369	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 7370	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 7371	Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
|- ( 1o ⊔ 1o ) ~~ 2o
Theorem	infpwfidom 7372	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 7373	Lemma for exmidfodomr 7378. A variant of djur 7232. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( ph -> A C_ 1o )   &    |- ( ph -> B e. ( A ⊔ 1o ) )   =>    |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
Theorem	exmidfodomrlemreseldju 7374	Lemma for exmidfodomrlemrALT 7377. A variant of eldju 7231. (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 7375*	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 7376*	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 7377*	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 7376. In particular, this proof uses eldju 7231 instead of djur 7232 and avoids djulclb 7218. (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 7378*	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 7379	Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( X e. AC A -> A e. _V )
Theorem	acneq 7380	Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( A = C -> AC A = AC C )
Theorem	isacnm 7381*	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 7382	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 7383	Formula for an abbreviation of the axiom of choice.
wff CHOICE
Definition	df-ac 7384*	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 4628 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 7385*	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 ) )
Theorem	exmidaclem 7386*	Lemma for exmidac 7387. 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 )
Theorem	exmidac 7387	The axiom of choice implies excluded middle. See acexmid 5999 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
Theorem	endjudisj 7388	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 ) )
Theorem	djuen 7389	Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )
Theorem	djuenun 7390	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 ) )
Theorem	dju1en 7391	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 )
Theorem	dju0en 7392	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 )
Theorem	xp2dju 7393	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 )
Theorem	djucomen 7394	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 ) )
Theorem	djuassen 7395	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 ) ) )
Theorem	xpdjuen 7396	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 ) ) )
Theorem	djudoml 7397	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 ) )
Theorem	djudomr 7398	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
Theorem	exmidontriimlem1 7399	Lemma for exmidontriim 7403. A variation of r19.30dc 2678. (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 ) )
Theorem	exmidontriimlem2 7400*	Lemma for exmidontriim 7403. (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 ) )