P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nninfisollemne 7201*	Lemma for nninfisol 7203. 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 7202*	Lemma for nninfisol 7203. 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 7203*	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. ℕ∞ ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
2.6.38  Omniscient sets
Syntax	comni 7204	Extend class definition to include the class of omniscient sets.
class Omni
Definition	df-omni 7205*	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 7206*	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 7207*	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 7208	Lemma for enomni 7209. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
Theorem	enomni 7209	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 6492 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 7210	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 7211	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7212. (Contributed by Jim Kingdon, 29-Jun-2022.)
|- (EXMID -> A. x x e. Omni )
Theorem	exmidomni 7212	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 7213	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
|- (EXMID -> om e. Omni )
Theorem	fodjuomnilemdc 7214*	Lemma for fodjuomni 7219. 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 7215*	Lemma for fodjuomni 7219 and fodjumkv 7230. 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 7216*	Lemma for fodjuomni 7219 and fodjumkv 7230. 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 7217*	Lemma for fodjuomni 7219 and fodjumkv 7230. 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 7218*	Lemma for fodjuomni 7219. 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 7219*	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 7220*	The decidability condition in ctssdc 7183 is needed. More specifically, ctssdc 7183 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 7221	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7222*	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 7223*	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 7224*	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 7225*	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 7226	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 7227	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
|- (EXMID -> om e. Markov )
Theorem	mkvprop 7228*	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 7229*	Lemma for fodjumkv 7230. 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 7230*	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 7231	Lemma for enmkv 7232. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
Theorem	enmkv 7232	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 6492 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 7233	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7234*	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 7235*	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 7236*	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 7237	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 7238	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7237. There is an analogue in terms of analytic omniscience principles at tridceq 15750. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
Theorem	enwomnilem 7239	Lemma for enwomni 7240. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
Theorem	enwomni 7240	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 6492 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 7241*	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 7242*	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 7243*	Lemma for nninfwlpor 7244. 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 7244*	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 7245*	Lemma for nninfwlpoim 7248. (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 7246*	Lemma for nninfwlpoim 7248. (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 7247*	Lemma for nninfwlpoim 7248. (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	nninfwlpoim 7248*	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	nninfwlpo 7249*	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 7250	Extend class definition to include the cardinal size function.
class card
Definition	df-card 7251*	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 } )
Theorem	cardcl 7252*	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 7253	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 7254	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 7255	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( A e. On -> A e. dom card )
Theorem	cardval3ex 7256*	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 7257*	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 7258	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 7259	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
|- ( card ` (/) ) = (/)
Theorem	ficardon 7260	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 7261*	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 7262	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 7263	Lemma for pr2ne 7264. (Contributed by FL, 17-Aug-2008.)
|- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o )
Theorem	pr2ne 7264	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	exmidonfinlem 7265*	Lemma for exmidonfin 7266. (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 7266	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 6937 and nnon 4647. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
|- ( om = ( On i^i Fin ) -> EXMID )
Theorem	en2eleq 7267	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 7268	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 7269	Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
|- ( 1o ⊔ 1o ) ~~ 2o
Theorem	infpwfidom 7270	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 7271	Lemma for exmidfodomr 7276. A variant of djur 7139. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( ph -> A C_ 1o )   &    |- ( ph -> B e. ( A ⊔ 1o ) )   =>    |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
Theorem	exmidfodomrlemreseldju 7272	Lemma for exmidfodomrlemrALT 7275. A variant of eldju 7138. (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 7273*	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 7274*	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 7275*	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 7274. In particular, this proof uses eldju 7138 instead of djur 7139 and avoids djulclb 7125. (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 7276*	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
Syntax	wac 7277	Formula for an abbreviation of the axiom of choice.
wff CHOICE
Definition	df-ac 7278*	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 4574 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 7279*	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 7280*	Lemma for exmidac 7281. 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 7281	The axiom of choice implies excluded middle. See acexmid 5924 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 7282	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 7283	Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )
Theorem	djuenun 7284	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 7285	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 7286	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 7287	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 7288	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 7289	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 7290	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 7291	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 7292	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 7293	Lemma for exmidontriim 7297. A variation of r19.30dc 2644. (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 7294*	Lemma for exmidontriim 7297. (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 ) )
Theorem	exmidontriimlem3 7295*	Lemma for exmidontriim 7297. 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 ) )
Theorem	exmidontriimlem4 7296*	Lemma for exmidontriim 7297. The induction step for the induction on A. (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 e. B \/ A = B \/ B e. A ) )
Theorem	exmidontriim 7297*	Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2.6.45  Excluded middle and the power set of a singleton
Theorem	pw1on 7298	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
Theorem	pw1dom2 7299	The power set of 1o dominates 2o. Also see pwpw0ss 3835 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
|- 2o ~<_ ~P 1o
Theorem	pw1ne0 7300	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)