mmtheorems64 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6301-6400 - Page 64 of 195

Type	Label	Description
Statement
Theorem	alephsucpw2 6301	The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous.) The transposed form alephsucpw 6511 cannot be proven without the AC, and is in fact equlvalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
|- -. ~P(aleph` A) ~< (aleph` suc A)
Cardinal number arithmetic
Syntax	ccda 6302	Extend class definition to include cardinal number addition.
class +c
Definition	df-cda 6303	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdavali 6305 for its value and a description.
|- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
Theorem	cdaval 6304	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.
|- ((A e. V /\ B e. W) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
Theorem	cdavali 6305	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 6470, carddom 6476, and cardsdom 6477. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.
|- A e. _V   &   |- B e. _V   =>   |- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))
Theorem	uncdadom 6306	Cardinal addition dominates union.
|- A e. _V   &   |- B e. _V   =>   |- (A u. B) ~<_ (A +c B)
Theorem	cdaun 6307	Cardinal addition is equinumerous to union for disjoint sets.
|- A e. _V   &   |- B e. _V   =>   |- ((A i^i B) = (/) -> (A +c B) ~~ (A u. B))
Theorem	cda1en 6308	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Revised by Mario Carneiro, 21-Feb-2013.)
|- A e. _V   =>   |- (-. A e. A -> (A +c 1o) ~~ suc A)
Theorem	cdaung 6309	Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Paul Chapman, 5-Jun-2009.)
|- ((A e. V /\ B e. W /\ (A i^i B) = (/)) -> (A +c B) ~~ (A u. B))
Theorem	pm110.643 6310	1+1=2 for cardinal number addition. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Unlike us, Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 6160), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 5930. The comment for cdavali 6305 explains why we use ~~ instead of =.
|- (1o +c 1o) ~~ 2o
Theorem	cdaen 6311	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   =>   |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))
Theorem	cdaeng 6312	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by Paul Chapman, 5-Jun-2009.)
|- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z) /\ (A ~~ B /\ C ~~ D)) -> (A +c C) ~~ (B +c D))
Theorem	cda0en 6313	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143.
|- A e. _V   =>   |- (A +c (/)) ~~ A
Theorem	xp1en 6314	One times a cardinal number.
|- A e. _V   =>   |- (A X. 1o) ~~ A
Theorem	xp2cda 6315	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
|- A e. _V   =>   |- (A X. 2o) = (A +c A)
Theorem	cdacomen 6316	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. _V   &   |- B e. _V   =>   |- (A +c B) ~~ (B +c A)
Theorem	cdaassen 6317	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   =>   |- ((A +c B) +c C) ~~ (A +c (B +c C))
Theorem	xpcdaen 6318	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   =>   |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))
Theorem	mapcdaen 6319	Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   =>   |- (A ^m (B +c C)) ~~ ((A ^m B) X. (A ^m C))
Theorem	cdadom1 6320	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   =>   |- (A ~<_ B -> (A +c C) ~<_ (B +c C))
Theorem	cdadom2 6321	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   =>   |- (A ~<_ B -> (C +c A) ~<_ (C +c B))
Theorem	cdadom3 6322	A set is dominated by its cardinal sum with another.
|- A e. _V   &   |- B e. _V   =>   |- A ~<_ (A +c B)
Theorem	cdafi 6323	The cardinal sum of two finite sets is finite.
|- ((A ~< om /\ B ~< om) -> (A +c B) ~< om)
Theorem	cdainflem 6324	Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.)
|- ((A u. B) ~~ om -> (A ~~ om \/ B ~~ om))
Theorem	cdainf 6325	A set is infinite iff the cardinal sum with itself is infinite. (Revised by Mario Carneiro, 11-Feb-2013.)
|- A e. _V   =>   |- (om ~<_ A <-> om ~<_ (A +c A))
Theorem	onacda 6326	The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.)
|- ((A e. On /\ B e. On) -> (A +o B) ~~ (A +c B))
Theorem	cardacda 6327	The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.)
|- (((A e. _V /\ B e. _V) /\ (E.x e. On x ~~ A /\ E.x e. On x ~~ B)) -> (A +c B) ~~ ((card` A) +o (card` B)))
Theorem	nnacda 6328	The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
|- ((A e. om /\ B e. om) -> (card` (A +c B)) = (A +o B))
Theorem	nnaun 6329	The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
|- ((A e. Fin /\ B e. Fin /\ (A i^i B) = (/)) -> (card` (A u. B)) = ((card` A) +o (card` B)))
Theorem	nnaun2 6330	The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
|- ((A e. Fin /\ B e. Fin) -> (card` (A u. B)) C_ ((card` A) +o (card` B)))
Theorem	pwsdompw 6331	Lemma for domtriom 6367. This is the equinumerosity version of the algebraic identity sum_k e. n(2^k) = (2^n) - 1. (Contributed by Mario Carneiro, 7-Feb-2013.) [Auxiliary lemma - not displayed.]
Cofinality (without Axiom of Choice)
Theorem	cflem 6332	A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set A.
|- (A e. V -> E.xE.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)))
Theorem	cfval 6333	Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A, there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is.
|- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
Theorem	cffnon 6334	Cofinality is a function on the class of ordinal numbers.
|- cf Fn On
Theorem	cfub 6335	An upper bound on cofinality.
|- (cf` A) C_ |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A C_ U.y))}
Theorem	cflim 6336	Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257.
|- ((A e. B /\ Lim A) -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A = U.y))})
Theorem	cf0 6337	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.
|- (cf` (/)) = (/)
Theorem	cardcf 6338	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.
|- (card` (cf` A)) = (cf` A)
Theorem	cflecard 6339	Cofinality is bounded by the cardinality of its argument.
|- (cf` A) C_ (card` A)
Theorem	cfle 6340	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.
|- (cf` A) C_ A
Theorem	cfon 6341	The cofinality of any set is an ordinal (although it only makes sense when A is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)
|- (cf` A) e. On
Theorem	cfeq0 6342	Only the ordinal zero has cofinality zero. (Revised by Mario Carneiro, 12-Feb-2013.)
|- (A e. On -> ((cf` A) = (/) <-> A = (/)))
Theorem	cfsuc 6343	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Revised by Mario Carneiro, 12-Feb-2013.)
|- (A e. On -> (cf` suc A) = 1o)
Theorem	cfom 6344	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Revised by Mario Carneiro, 12-Feb-2013.)
|- (cf` om) = om
Theorem	cff1 6345	There is always a map from (cf` A) to A (this is a stronger condition than the definition, which only presupposes a map from some y ~~ (cf` A). (Contributed by Mario Carneiro, 28-Feb-2013.)
|- (A e. On -> E.f(f:(cf` A)-1-1->A /\ A.z e. A E.w e. (cf` A)z C_ (f` w)))
Theorem	cfflb 6346	If there is a cofinal map from B to A, then B is at least (cf` A). This theorem and cff1 6345 motivate the picture of (cf` A) as the greatest lower bound of the domain of cofinal maps into A. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ((A e. On /\ B e. On) -> (E.f(f:B-->A /\ A.z e. A E.w e. B z C_ (f` w)) -> (cf` A) C_ B))
Theorem	cfval2 6347	Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- (A e. On -> (cf` A) = |^|_x e. {x e. ~PA | A.z e. A E.w e. x z C_ w} (card` x))
Theorem	coflim 6348	A simpler expression for the cofinality predicate, at a limit ordinal.
|- ((Lim A /\ B C_ A) -> (U.B = A <-> A.x e. A E.y e. B x C_ y))
Theorem	cflim3 6349	Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- A e. _V   =>   |- (Lim A -> (cf` A) = |^|_x e. {x e. ~PA | U.x = A} (card` x))
Theorem	cflim2 6350	The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- A e. _V   =>   |- (Lim A <-> Lim (cf` A))
Theorem	cofsmo 6351	Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by by Mario Carneiro, 20-Mar-2013.)
|- C = {y e. B | A.w e. y (f` w) e. (f` y)}   &   |- K = |^|{x e. B | z C_ (f` x)}   =>   |- ((Ord A /\ B e. On) -> (E.f(f:B-->A /\ A.z e. A E.w e. B z C_ (f` w)) -> E.x e. suc BE.g(g:x-->A /\ Smo g /\ A.z e. A E.v e. x z C_ (g` v))))
Theorem	cfsmolem 6352	Lemma for cfsmo 6353. [Auxiliary lemma - not displayed.]
Theorem	cfsmo 6353	The map in cff1 6345 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- (A e. On -> E.f(f:(cf` A)-->A /\ Smo f /\ A.z e. A E.w e. (cf` A)z C_ (f` w)))
Theorem	cfcoflem 6354	Lemma for cfcof 6356, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) [Auxiliary lemma - not displayed.]
Theorem	coftr 6355	If there is a cofinal map from B to A and another from C to A, then there is also a cofinal map from C to B. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 6356. (Contributed by Mario Carneiro, 13-Mar-2013.)
|- H = (t e. C |-> |^|{n e. B | (g` t) C_ (f` n)})   =>   |- (E.f(f:B-->A /\ Smo f /\ A.x e. A E.y e. B x C_ (f` y)) -> (E.g(g:C-->A /\ A.z e. A E.w e. C z C_ (g` w)) -> E.h(h:C-->B /\ A.s e. B E.w e. C s C_ (h` w))))
Theorem	cfcof 6356	If there is a cofinal map from A to B, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof (A, B) and defines our cf(B) as the minimum B such that cof (A, B). (Contributed by Mario Carneiro, 20-Mar-2013.)
|- ((A e. On /\ B e. On) -> (E.f(f:B-->A /\ Smo f /\ A.z e. A E.w e. B z C_ (f` w)) -> (cf` A) = (cf` B)))
Theorem	cfidm 6357	The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.)
|- (cf` (cf` A)) = (cf` A)
Theorem	alephsing 6358	The cofinality of a limit aleph is the same as the cofinality of its argument, so if (aleph` A) < A, then (aleph` A) is singular. Conversely, if (aleph` A) is regular (i.e. weakly inaccessible), then (aleph` A) = A, so A has to be rather large (see alephfp 6297). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
|- (Lim A -> (cf` (aleph` A)) = (cf` A))
ZFC Set Theory - add Countable Choice and Dependent Choice
Axiom	ax-cc 6359	The axiom of countable choice (CC). It is clearly a special case of ac5 6393, but is weak enough that it can be proven using DC (see axcc 6384). It is, however, strictly stronger than ZF and cannot be proven in ZF. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- (x ~~ om -> E.fA.z e. x (z =/= (/) -> (f` z) e. z))
Theorem	axcc2lem 6360	Lemma for axcc2 6361. [Auxiliary lemma - not displayed.]
Theorem	axcc2 6361	A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
|- E.g(g Fn om /\ A.n e. om ((F` n) =/= (/) -> (g` n) e. (F` n)))
Theorem	axcc3 6362	A possibly more useful version of ax-cc 6359 using sequences F(n) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
|- F e. _V   &   |- N ~~ om   =>   |- E.f(f Fn N /\ A.n e. N (F =/= (/) -> (f` n) e. F))
Theorem	axcc4 6363	A version of axcc3OLD 6364 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
|- A e. _V   &   |- N ~~ om   &   |- (x = (f` n) -> (ph <-> ps))   =>   |- (A.n e. N E.x e. A ph -> E.f(f:N-->A /\ A.n e. N ps))
Theorem	axcc3OLD 6364	A possibly more useful version of ax-cc using sequences F(n) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
|- E.g(g Fn om /\ A.n e. om (F =/= (/) -> (g` n) e. F))
Theorem	domtriomlem 6365	Lemma for domtriom 6367. [Auxiliary lemma - not displayed.]
Theorem	domtriomlemOLD 6366	Lemma for domtriom 6367. [Auxiliary lemma - not displayed.]
Theorem	domtriom 6367	Trichotomy of equinumerosity for om, proven using CC. (Revised by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>   |- (om ~<_ A <-> -. A ~< om)
Theorem	domtriomOLD 6368	Trichotomy of equinumerosity for om, proven using CC. (Revised by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>   |- (om ~<_ A <-> -. A ~< om)
Theorem	domtriomlemOLDOLD 6369	Lemma for domtriomOLDOLD 6370. [Auxiliary lemma - not displayed.]
Theorem	domtriomOLDOLD 6370	Trichotomy of equinumerosity for om, proven using CC. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. B   =>   |- (om ~<_ A <-> -. A ~< om)
Theorem	dominf 6371	A nonempty set that is a subset of its union is infinite. Equivalently, there are no Dedekind-finite sets. This version is proved from ax-cc 6359. See dominfac 6522 for a version proved from ax-ac 6385. The axiom of Regularity is used for this proof, via inf3lem6 6000, and its use is necessary: otherwise the set A = {A} or A = {(/), A} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>   |- ((A =/= (/) /\ A C_ U.A) -> om ~<_ A)
Axiom	ax-dc 6372	Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. This axiom is redundant in ZFC; see axdc 6447. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.)
|- ((E.yE.z yxz /\ ran x C_ dom x) -> E.fA.n e. om (f` n)x(f` suc n))
Theorem	dcomex 6373	The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
|- om e. _V
Theorem	axdc2lem 6374	Lemma for axdc2 6375. We construct a relation R based on F such that xRy iff y e. (F` x), and show that the "function" described by ax-dc 6372 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). [Auxiliary lemma - not displayed.]
Theorem	axdc2 6375	An apparent strengthening of ax-dc 6372 (but derived from it) which shows that there is a denumerable sequence g for any function that maps elements of a set A to nonempty subsets of A such that g(x + 1) e. F(g(x)) for all x e. om. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
|- A e. _V   =>   |- ((A =/= (/) /\ F:A-->(~PA \ {(/)})) -> E.g(g:om-->A /\ A.k e. om (g` suc k) e. (F` (g` k))))
Theorem	axdc3lem 6376	The class S of finite approximations to the DC sequence is a set. (We derive here the stronger statement that S is a subset of a specific set, namely ~P(om X. A).)
|- A e. _V   &   |- S = {s | E.n e. om (s:suc n-->A /\ (s` (/)) = C /\ A.k e. n (s` suc k) e. (F` (s` k)))}   =>   |- S e. _V
Theorem	axdc3lem2 6377	Lemma for acdc3 9271. We have constructed a "candidate set" S, which consists of all finite sequences s that satisfy our property of interest, namely s(x + 1) e. F(s(x)) on its domain, but with the added constraint that s(0) = C. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 6372 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (h` n):m-->A (for some integer m). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 6372 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence h, we can construct the sequence g that we are after. [Auxiliary lemma - not displayed.]
Theorem	axdc3lem3 6378	Simple substitution lemma for axdc3 6380.
|- A e. _V   &   |- S = {s | E.n e. om (s:suc n-->A /\ (s` (/)) = C /\ A.k e. n (s` suc k) e. (F` (s` k)))}   &   |- B e. _V   =>   |- (B e. S <-> E.m e. om (B:suc m-->A /\ (B` (/)) = C /\ A.k e. m (B` suc k) e. (F` (B` k))))
Theorem	axdc3lem4 6379	Lemma for acdc3 9271. We have constructed a "candidate set" S, which consists of all finite sequences s that satisfy our property of interest, namely s(x + 1) e. F(s(x)) on its domain, but with the added constraint that s(0) = C. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 6372 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (h` n):m-->A (for some integer m). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 6372 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that S is nonempty, and that G always maps to a nonempty subset of S, so that we can apply axdc2 6375. See axdc3lem2 6377 for the rest of the proof. [Auxiliary lemma - not displayed.]
Theorem	axdc3 6380	Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value C. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.)
|- A e. _V   =>   |- ((C e. A /\ F:A-->(~PA \ {(/)})) -> E.g(g:om-->A /\ (g` (/)) = C /\ A.k e. om (g` suc k) e. (F` (g` k))))
Theorem	axdc4lem 6381	Lemma for axdc4 6382. [Auxiliary lemma - not displayed.]
Theorem	axdc4 6382	A more general version of axdc3 6380 that allows the function F to vary with k. (Contributed by Mario Carneiro, 31-Jan-2013.)
|- A e. _V   =>   |- ((C e. A /\ F:(om X. A)-->(~PA \ {(/)})) -> E.g(g:om-->A /\ (g` (/)) = C /\ A.k e. om (g` suc k) e. (kF(g` k))))
Theorem	axcclem 6383	Lemma for axcc 6384. [Auxiliary lemma - not displayed.]
Theorem	axcc 6384	Although CC can be proven trivially using ac5 6393, we prove it here using DC. (Contributed by Mario Carneiro, 2-Feb-2013.)
|- (x ~~ om -> E.fA.z e. x (z =/= (/) -> (f` z) e. z))
ZFC Set Theory - add the Axiom of Choice
Introduce the Axiom of Choice
Axiom	ax-ac 6385	Axiom of Choice. The Axiom of Choice (AC) is usually considered an extension of ZF set theory rather than a proper part of it. It is sometimes considered philosophically controversial because it asserts the existence of a set without telling us what the set is. ZF set theory that includes AC is called ZFC. The unpublished version given here says that given any set x, there exists a y that is a collection of unordered pairs, one pair for each non-empty member of x. One entry in the pair is the member of x, and the other entry is some arbitrary member of that member of x. See the rewritten version ac3 6388 for a more detailed explanation. Theorem ac2 6387 shows an equivalent written compactly with restricted quantifiers. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 6428 is slightly shorter when the biconditional of ax-ac 6385 is expanded into implication and negation. Standard textbook versions of AC are derived as ac8 6409, ac5 6393, and ac7 6389. The Axiom of Regularity ax-reg 5972 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 6261. Equivalents to AC are the well-ordering theorem weth 6433 and Zorn's lemma zorn 6444. See ac4 6391 for comments about stronger versions of AC.
|- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
Theorem	zfac 6386	Axiom of Choice expressed with fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 6385.
|- E.xA.yA.z((y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w))
Theorem	ac2 6387	Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 6388 is easier to understand.) Note: aceq0 6249 shows the logical equivalence to ax-ac 6385.
|- E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u)
Theorem	ac3 6388	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 6385 can be established by chaining aceq0 6249 and aceq2 6250. A standard textbook version of AC is derived from this one in aceq6a 6260, and this version of AC is derived from the textbook version in aceq6b 6261. The following sketch will help you understand this version of the axiom. Given any set x, the axiom says that there exists a y that is a collection of unordered pairs, one pair for each non-empty member of x. One entry in the pair is the member of x, and the other entry is some arbitrary member of that member of x. Using the Axiom of Regularity, we can show that y is really a set of ordered pairs, very similar to the ordered pair construction opthreg 5985. The key theorem for this (used in the proof of aceq6b 6261) is preleq 5984. With this modified definition of ordered pair, it can be seen that y is actually a choice function on the members of x. For example, suppose x = {{1, 2}, {1, 3}, {2, 3}}. Take y = {{{1, 2}, 1}, {{1, 3}, 1}, {{2, 3}, 2}}. For the member (of x) z = {1, 2}, the only assignment to w and v that satisfies the axiom is w = 1 and v = {{1, 2}, 1}, so there is exactly one w as required. We verify the other two members of x similarly. Thus y satisfies the axiom. Using our modified ordered pair definition, it is easy to see that y is the choice function {<.{1, 2}, 1>., <.{1, 3}, 1>., <.{2, 3}, 2>.}. Of course other choices for y will also satisfy the axiom, for example y = {{{1, 2}, 2}, {{1, 3}, 1}, {{2, 3}, 3}}. What AC tells us is that there exists at least one such y, but it doesn't tell us which one.
|- E.yA.z e. x (z =/= (/) -> E!w e. z E.v e. y (z e. v /\ w e. v))
Theorem	ac7 6389	An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49.
|- E.f(f C_ x /\ f Fn dom x)
Theorem	ac7g 6390	An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49.
|- (R e. A -> E.f(f C_ R /\ f Fn dom R))
Theorem	ac4 6391	Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 6393, derived from this one, shows that this form of the axiom does imply that at least one such set f whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83. Takeuti and Zaring call this "weak choice" in contrast to "strong choice" E.FA.z(z =/= (/) -> (F` z) e. z), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable F and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971). Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 6408.
|- E.fA.z e. x (z =/= (/) -> (f` z) e. z)
Theorem	ac4c 6392	Equivalent of Axiom of Choice (class version)
|- A e. _V   =>   |- E.fA.x e. A (x =/= (/) -> (f` x) e. x)
Theorem	ac5 6393	An Axiom of Choice equivalent: there exists a function f (called a choice function) with domain A that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that f be a function is not necessary; see ac4 6391.
|- A e. _V   =>   |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))
Theorem	ac5b 6394	Equivalent of Axiom of Choice.
|- A e. _V   =>   |- (A.x e. A x =/= (/) -> E.f(f:A-->U.A /\ A.x e. A (f` x) e. x))
Theorem	ac6lem 6395	Lemma for ac6 6396. [Auxiliary lemma - not displayed.]
Theorem	ac6 6396	Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set B, where ph depends on x (the natural number) and y (to specify a member of B). A stronger version of this theorem, ac6s 6400, allows B to be a proper class.
|- A e. _V   &   |- B e. _V   &   |- (y = (f` x) -> (ph <-> ps))   =>   |- (A.x e. A E.y e. B ph -> E.f(f:A-->B /\ A.x e. A ps))
Theorem	ac6c4 6397	Equivalent of Axiom of Choice. B is a collection B(x) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
|- A e. _V   &   |- B e. _V   =>   |- (A.x e. A B =/= (/) -> E.f(f Fn A /\ A.x e. A (f` x) e. B))
Theorem	ac6c5 6398	Equivalent of Axiom of Choice. B is a collection B(x) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
|- A e. _V   &   |- B e. _V   =>   |- (A.x e. A B =/= (/) -> E.fA.x e. A (f` x) e. B)
Theorem	ac9 6399	An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)
|- A e. _V   &   |- B e. _V   =>   |- (A.x e. A B =/= (/) <-> X_x e. A B =/= (/))
Theorem	ac6s 6400	Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 6158, we derive this strong version of ac6 6396 that doesn't require B to be a set.
|- A e. _V   &   |- (y = (f` x) -> (ph <-> ps))   =>   |- (A.x e. A E.y e. B ph -> E.f(f:A-->B /\ A.x e. A ps))