mmtheorems48 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4701-4800 - Page 48 of 107

Type	Label	Description
Statement
Theorem	rankxplim2 4701	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 4702	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 4703	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 4700 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 4704	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 4705	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 4706	Theorem scheme version of scottex 4704. 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 4707	Theorem scheme version of scott0 4705. 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 4708	Lemma for the Collection Principle cp 4710.
Theorem	cplem2 4709	Lemma for the Collection Principle cp 4710.
Theorem	cp 4710	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 4704 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 4711	A very strong generalization of the Axiom of Replacement (compare zfrep6 3612), derived from the Collection Principle cp 4710. 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 4712	A variant of the Boundedness Axiom bnd 4711 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 4713	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 4714	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 4819). 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 4713 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 4715	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 4716	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 1295 and weth 4775, using scottexs 4706 to establish the existence of A. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 4715.
|- 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 4717	Equivalence of two versions of the Axiom of Choice ax-ac 4732. 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 4718	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 4732.
|- (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 4719	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 4720	Lemma for aceq3 4721.
Theorem	aceq3 4721	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 4722	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 4723	Lemma for aceq5 4728.
Theorem	aceq5lem2 4724	Lemma for aceq5 4728.
Theorem	aceq5lem3 4725	Lemma for aceq5 4728.
Theorem	aceq5lem4 4726	Lemma for aceq5 4728.
Theorem	aceq5lem5 4727	Lemma for aceq5 4728.
Theorem	aceq5 4728	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 4729	Our Axiom of Choice (in the form of ac3 4735) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See aceq6b 4730 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 4730	Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 4735). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 4586 and preleq 4591 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 4729.)
|- (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 4731	Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 4734). 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 4732	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 4735 for a more detailed explanation. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 4770 is slightly shorter when the biconditional of ax-ac 4732 is expanded into implication and negation. Standard textbook versions of AC are derived as ac8 4751, ac5 4740, and ac7 4736. The Axiom of Regularity ax-reg 4581 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 4730. Equivalents to AC are the well-ordering theorem weth 4775 and Zorn's lemma zorn 4785. See ac4 4738 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	axac 4733	Axiom of Choice expressed with fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 4732.
|- 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 4734	Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single conjunction. (If you want to figure it out, the rewritten equivalent ac3 4735 is easier to understand.) Note: aceq0 4718 shows the logical equivalence to ax-ac 4732.
|- 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 4735	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 4732 can be established by chaining aceq0 4718 and aceq2 4719. A standard textbook version of AC is derived from this one in aceq6a 4729, and this version of AC is derived from the textbook version in aceq6b 4730. 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 4592. The key theorem for this (used in the proof of aceq6b 4730) is preleq 4591. 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 4736	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 4737	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 4738	Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 4740, 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.