mmtheorems49 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4801-4900 - Page 49 of 123

Type	Label	Description
Statement
Theorem	r1ord 4801	Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77.
|- (B e. On -> (A e. B -> (R1` A) e. (R1` B)))
Theorem	r1ord2 4802	Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77.
|- (B e. On -> (A e. B -> (R1` A) (_ (R1` B)))
Theorem	r1ord3 4803	Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478.
|- ((A e. On /\ B e. On) -> (A (_ B -> (R1` A) (_ (R1` B)))
Theorem	r1val1 4804	The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202.
|- (A e. On -> (R1` A) = U_x e. A P~(R1` x))
Theorem	tz9.12lem1 4805	Lemma for tz9.12 4808.
Theorem	tz9.12lem2 4806	Lemma for tz9.12 4808.
Theorem	tz9.12lem3 4807	Lemma for tz9.12 4808.
Theorem	tz9.12 4808	A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 4805 through tz9.12lem3 4807.
|- A e. V   =>   |- (A.x e. A E.y e. On x e. (R1` y) -> E.y e. On A e. (R1` y))
Theorem	tz9.13 4809	Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78.
|- A e. V   =>   |- E.x e. On A e. (R1` x)
Theorem	tz9.13g 4810	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 4809 expresses the class existence requirement as an antecedent.
|- (A e. B -> E.x e. On A e. (R1` x))
Theorem	rankwflem 4811	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 4810 is useful in proofs of theorems about the rank function.
|- (A e. B -> E.x e. On A e. (R1` suc x))
Theorem	jech9.3 4812	Every set belongs to some value of the cumulative hierarchy of sets function R1, i.e. the indexed union of all values of R1 is the universe. Lemma 9.3 of [Jech] p. 71.
|- U_x e. On (R1` x) = V
Theorem	unir1 4813	The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281.
|- U.(R1"On) = V
Theorem	rankval 4814	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition).
|- A e. V   =>   |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}
Theorem	rankvalg 4815	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 4814 expresses the class existence requirement as an antecedent instead of a hypothesis.
|- (A e. B -> (rank` A) = |^|{x e. On | A e. (R1` suc x)})
Theorem	rankval2 4816	Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478.
|- (A e. B -> (rank` A) = |^|{x e. On | A (_ (R1` x)})
Theorem	rankon 4817	The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79.
|- (rank` A) e. On
Theorem	rankid 4818	Identity law for the rank function.
|- A e. V   =>   |- A e. (R1` suc (rank` A))
Theorem	rankr1lem 4819	Lemma for rankr1 4820.
Theorem	rankr1 4820	A relationship between the rank function and the cumulative hierarchy of sets function R1. Proposition 9.15(2) of [TakeutiZaring] p. 79.
|- A e. V   =>   |- (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B)))
Theorem	rankr1g 4821	A relationship between the rank function and the cumulative hierarchy of sets function R1. Proposition 9.15(2) of [TakeutiZaring] p. 79.
|- (A e. C -> (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B))))
Theorem	ssrankr1 4822	A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets R1. Proposition 9.15(3) of [TakeutiZaring] p. 79.
|- A e. V   =>   |- (B e. On -> (B (_ (rank` A) <-> -. A e. (R1` B)))
Theorem	rankr1a 4823	A relationship between rank and R1, clearly equivalent to ssrankr1 4822 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 4845 for the subset verion. (Contributed by Raph Levien, 29-May-2004.)
|- A e. V   =>   |- (B e. On -> (A e. (R1` B) <-> (rank` A) e. B))
Theorem	r1val2 4824	The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113.
|- (A e. On -> (R1` A) = {x | (rank` x) e. A})
Theorem	r1val3 4825	The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113.
|- (A e. On -> (R1` A) = U_x e. A P~{y | (rank` y) e. x})
Theorem	rankel 4826	The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79.
|- B e. V   =>   |- (A e. B -> (rank` A) e. (rank` B))
Theorem	rankval3 4827	The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79.
|- A e. V   =>   |- (rank` A) = |^|{x e. On | A.y e. A (rank` y) e. x}
Theorem	bndrank 4828	Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80.
|- (E.x e. On A.y e. A (rank` y) (_ x -> A e. V)
Theorem	unbndrank 4829	The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80.
|- (-. A e. V -> A.x e. On E.y e. A x e. (rank` y))
Theorem	rankpw 4830	The rank of a power set. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   =>   |- (rank` P~A) = suc (rank` A)
Theorem	ranklim 4831	The rank of a set belongs to a limit ordinal iff the rank of its power set does.
|- (Lim B -> ((rank` A) e. B <-> (rank` P~A) e. B))
Theorem	r1pw 4832	A stronger property of R1 than rankpw 4830. The latter merely proves that R1 of the successor is a power set, but here we prove that if A is in the cumulative hierarchy, then P~A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.)
|- (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B)))
Theorem	r1pwcl 4833	The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.)
|- (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B)))
Theorem	rankss 4834	The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80.
|- B e. V   =>   |- (A (_ B -> (rank` A) (_ (rank` B))
Theorem	ranksn 4835	The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112.
|- A e. V   =>   |- (rank` {A}) = suc (rank` A)
Theorem	rankuni2 4836	The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112.
|- A e. V   =>   |- (rank` U.A) = U_x e. A (rank` x)
Theorem	rankun 4837	The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112.
|- A e. V   &   |- B e. V   =>   |- (rank` (A u. B)) = ((rank` A) u. (rank` B))
Theorem	rankpr 4838	The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   &   |- B e. V   =>   |- (rank` {A, B}) = suc ((rank` A) u. (rank` B))
Theorem	rankop 4839	The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107.
|- A e. V   &   |- B e. V   =>   |- (rank` <.A, B>.) = suc suc ((rank` A) u. (rank` B))
Theorem	r1rankid 4840	Any set is a subset of the hierarchy of its rank.
|- (A e. B -> A (_ (R1` (rank` A)))
Theorem	rankonid 4841	The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse.
|- (A e. On <-> (rank` A) = A)
Theorem	rankeq0 4842	A set is empty iff its rank is empty.
|- A e. V   =>   |- (A = (/) <-> (rank` A) = (/))
Theorem	rankr1id 4843	The rank of the hierarchy of an ordinal number is itself.
|- (A e. On <-> (rank` (R1` A)) = A)
Theorem	rankuni 4844	The rank of a union. Part of Exercise 4 of [Kunen] p. 107.
|- (rank` U.A) = U.(rank` A)
Theorem	rankr1b 4845	A relationship between rank and R1. See rankr1a 4823 for the membership version.
|- A e. V   =>   |- (B e. On -> (A (_ (R1` B) <-> (rank` A) (_ B))
Theorem	ranksuc 4846	The rank of a successor.
|- A e. V   =>   |- (rank` suc A) = suc (rank` A)
Theorem	rankuniss 4847	Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   =>   |- (rank` U.A) (_ (rank` A)
Theorem	rankval4 4848	The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204.
|- A e. V   =>   |- (rank` A) = U_x e. A suc (rank` x)
Theorem	rankbnd 4849	The rank of a set is bounded by a bound for the successor of its members.
|- A e. V   =>   |- (A.x e. A suc (rank` x) (_ B <-> (rank` A) (_ B)
Theorem	rankbnd2 4850	The rank of a set is bounded by the successor of a bound for its members.
|- A e. V   =>   |- (B e. On -> (A.x e. A (rank` x) (_ B <-> (rank` A) (_ suc B))
Theorem	rankc1 4851	A relationship that can be used for computation of rank.
|- A e. V   =>   |- (A.x e. A (rank` x) e. (rank` U.A) <-> (rank` A) = (rank` U.A))
Theorem	rankc2 4852	A relationship that can be used for computation of rank.
|- A e. V   =>   |- (E.x e. A (rank` x) = (rank` U.A) -> (rank` A) = suc (rank` U.A))
Theorem	rankelun 4853	Rank membership is inherited by union.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. (rank` (C u. D)))
Theorem	rankelpr 4854	Rank membership is inherited by unordered pairs.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` {A, B}) e. (rank` {C, D}))
Theorem	rankelop 4855	Rank membership is inherited by ordered pairs.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` <.A, B>.) e. (rank` <.C, D>.))
Theorem	rankxpl 4856	A lower bound on the rank of a cross product.
|- A e. V   &   |- B e. V   =>   |- ((A X. B) =/= (/) -> (rank` (A u. B)) (_ (rank` (A X. B)))
Theorem	rankxpu 4857	An upper bound on the rank of a cross product.
|- A e. V   &   |- B e. V   =>   |- (rank` (A X. B)) (_ suc suc (rank` (A u. B))
Theorem	rankxplim 4858	The rank of a cross product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 4861 for the successor case.
|- A e. V   &   |- B e. V   =>   |- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = (rank` (A u. B)))
Theorem	rankxplim2 4859	If the rank of a cross product is a limit ordinal, so is the rank of the union of its arguments.
|- A e. V   &   |- B e. V   =>   |- (Lim (rank` (A X. B)) -> Lim (rank` (A u. B)))
Theorem	rankxplim3 4860	The rank of a cross product is a limit ordinal iff its union is.
|- A e. V   &   |- B e. V   =>   |- (Lim (rank` (A X. B)) <-> Lim U.(rank` (A X. B)))
Theorem	rankxpsuc 4861	The rank of a cross product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 4858 for the limit ordinal case.
|- A e. V   &   |- B e. V   =>   |- (((rank` (A u. B)) = suc C /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = suc suc (rank` (A u. B)))
Scott's trick; collection principle; Hilbert's epsilon
Theorem	scottex 4862	Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set.
|- {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V
Theorem	scott0 4863	Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. A is empty).
|- (A = (/) <-> {x e. A | A.y e. A (rank` x) (_ (rank` y)} = (/))
Theorem	scottexs 4864	Theorem scheme version of scottex 4862. The collection of all x of minimum rank such that ph(x) is true, is a set.
|- {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} e. V
Theorem	scott0s 4865	Theorem scheme version of scott0 4863. The collection of all x of minimum rank such that ph(x) is true, is not empty iff there is an x such that ph(x) holds.
|- (E.xph <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} =/= (/))
Theorem	cplem1 4866	Lemma for the Collection Principle cp 4868.
Theorem	cplem2 4867	-Lemma for the Collection Principle cp 4868.
|- A e. V   =>   |- E.yA.x e. A (B =/= (/) -> (B i^i y) =/= (/))
Theorem	cp 4868	Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 4862 that collapses a proper class into a set of minimum rank. The wff ph can be thought of as ph(x, y). Scheme "Collection Principle" of [Jech] p. 72.
|- E.wA.x e. z (E.yph -> E.y e. w ph)
Theorem	bnd 4869	A very strong generalization of the Axiom of Replacement (compare zfrep6 3721), derived from the Collection Principle cp 4868. Its strength lies in the rather profound fact that ph(x, y) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom.
|- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)
Theorem	bnd2 4870	A variant of the Boundedness Axiom bnd 4869 that picks a subset z out of a possibly proper class B in which a property is true.
|- A e. V   =>   |- (A.x e. A E.y e. B ph -> E.z(z (_ B /\ A.x e. A E.y e. z ph))
Theorem	kardex 4871	The collection of all sets equinumerous to a set A and having least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222.
|- {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))} e. V
Theorem	karden 4872	If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 4979). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 4871 justify the definition of kard. The restriction to least rank prevents the proper class that would result from {x | x ~~ A}.
|- A e. V   &   |- B e. V   &   |- C = {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))}   &   |- D = {x | (x ~~ B /\ A.y(y ~~ B -> (rank` x) (_ (rank` y)))}   =>   |- (C = D <-> A ~~ B)
Theorem	htalem 4873	Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional R We A antecedent. The element B is the epsilon that the theorem emulates.
Theorem	hta 4874	A ZFC emulation of Hilbert's transfinite axiom. The set B has the properties of Hilbert's epsilon, except that it also depends on a well-ordering R. This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See http://ghilbert.org/choice.txt and http://us.metamath.org/downloads/megillaward2004.pdf. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires R We A as an antecedent. Class A collects the sets of least rank for which ph(x) is true. Class B, which emulates the epsilon, is the minimum element in a well-ordering R on A. If a well-ordering R on A can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace R with a dummy set variable, say w, and attach w We A as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, B (which will have w as a free variable) will no longer be present, and we can eliminate w We A by applying 19.23aiv 1333 and weth 4933, using scottexs 4864 to establish the existence of A. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 4873.
|- A = {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))}   &   |- B = U.{x e. A | A.y e. A -. yRx}   =>   |- (R We A -> (ph -> [B / x]ph))
Axiom of Choice equivalents
Theorem	aceq1 4875	Equivalence of two versions of the Axiom of Choice ax-ac 4890. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables.
|- (E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u) <-> E.yA.zA.w((z e. w /\ w e. x) -> E.xA.z(E.x((z e. w /\ w e. x) /\ (z e. x /\ x e. y)) <-> z = x)))
Theorem	aceq0 4876	Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 4890.
|- (E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u) <-> 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	aceq2 4877	Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity.
|- (E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u) <-> E.yA.z e. x (z =/= (/) -> E!w e. z E.v e. y (z e. v /\ w e. v)))
Theorem	aceq3lem 4878	Lemma for aceq3 4879.
Theorem	aceq3 4879	Equivalence of two versions of the Axiom of Choice. The left-hand side is similar to the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC.
|- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z))
Theorem	aceq4 4880	Equivalence of two versions of the Axiom of Choice. The left-hand side is similar to the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC.
|- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.f(f Fn x /\ A.z e. x (z =/= (/) -> (f` z) e. z)))
Theorem	aceq5lem1 4881	Lemma for aceq5 4886.
Theorem	aceq5lem2 4882	Lemma for aceq5 4886.
Theorem	aceq5lem3 4883	Lemma for aceq5 4886.
Theorem	aceq5lem4 4884	Lemma for aceq5 4886.
Theorem	aceq5lem5 4885	Lemma for aceq5 4886.
Theorem	aceq5 4886	Equivalence of two versions of the Axiom of Choice. The left-hand side is similar to the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC.
|- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.x((A.z e. x z =/= (/) /\ A.z e. x A.w e. x (z =/= w -> (z i^i w) = (/))) -> E.yA.z e. x E!v v e. (z i^i y)))
Theorem	aceq6a 4887	Our Axiom of Choice (in the form of ac3 4893) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See aceq6b 4888 for the converse (which does use the Axiom of Regularity).
|- (A.xE.yA.z e. x (z =/= (/) -> E!w e. z E.v e. y (z e. v /\ w e. v)) -> A.xE.f(f (_ x /\ f Fn dom x))
Theorem	aceq6b 4888	Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 4893). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 4741 and preleq 4748 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see aceq6a 4887.)
|- (A.xE.f(f (_ x /\ f Fn dom x) -> A.xE.yA.z e. x (z =/= (/) -> E!w e. z E.v e. y (z e. v /\ w e. v)))
Theorem	aceq7 4889	Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 4892). The proof does not depend AC on but does depend on the Axiom of Regularity.
|- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u))
ZFC Set Theory - add the Axiom of Choice
Introduce the Axiom of Choice
Axiom	ax-ac 4890	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 4893 for a more detailed explanation. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 4928 is slightly shorter when the biconditional of ax-ac 4890 is expanded into implication and negation. Standard textbook versions of AC are derived as ac8 4909, ac5 4898, and ac7 4894. The Axiom of Regularity ax-reg 4736 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 4888. Equivalents to AC are the well-ordering theorem weth 4933 and Zorn's lemma zorn 4943. See ac4 4896 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 4891	Axiom of Choice expressed with fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 4890.
|- 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 4892	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 4893 is easier to understand.) Note: aceq0 4876 shows the logical equivalence to ax-ac 4890.
|- 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 4893	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 4890 can be established by chaining aceq0 4876 and aceq2 4877. A standard textbook version of AC is derived from this one in aceq6a 4887, and this version of AC is derived from the textbook version in aceq6b 4888. 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 4749. The key theorem for this (used in the proof of aceq6b 4888) is preleq 4748. 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 4894	An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49.
|- E.f(f (_ x /\ f Fn dom x)
Theorem	ac7g 4895	An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49.
|- (R e. A -> E.f(f (_ R /\ f Fn dom R))
Theorem	ac4 4896	Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 4898, 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 4908.
|- E.fA.z e. x (z =/= (/) -> (f` z) e. z)
Theorem	ac4c 4897	Equivalent of Axiom of Choice (class version)
|- A e. V   =>   |- E.fA.x e. A (x =/= (/) -> (f` x) e. x)
Theorem	ac5 4898	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 4896.
|- A e. V   =>   |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))
Theorem	ac5b 4899	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 4900	Lemma for ac6 4901.