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	scottexs 4701	Theorem scheme version of scottex 4699. 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 4702	Theorem scheme version of scott0 4700. 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 4703	Lemma for the Collection Principle cp 4705.
Theorem	cplem2 4704	Lemma for the Collection Principle cp 4705.
Theorem	cp 4705	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 4699 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 4706	A very strong generalization of the Axiom of Replacement (compare zfrep6 3609), derived from the Collection Principle cp 4705. 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 4707	A variant of the Boundedness Axiom bnd 4706 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 4708	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 4709	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 4814). 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 4708 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 4710	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 4711	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 1294 and weth 4770, using scottexs 4701 to establish the existence of A. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 4710.
|- 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 4712	Equivalence of two versions of the Axiom of Choice ax-ac 4727. 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 4713	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 4727.
|- (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 4714	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 4715	Lemma for aceq3 4716.
Theorem	aceq3 4716	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 4717	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 4718	Lemma for aceq5 4723.
Theorem	aceq5lem2 4719	Lemma for aceq5 4723.
Theorem	aceq5lem3 4720	Lemma for aceq5 4723.
Theorem	aceq5lem4 4721	Lemma for aceq5 4723.
Theorem	aceq5lem5 4722	Lemma for aceq5 4723.
Theorem	aceq5 4723	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 4724	Our Axiom of Choice (in the form of ac3 4730) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See aceq6b 4725 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 4725	Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 4730). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 4581 and preleq 4586 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 4724.)
|- (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 4726	Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 4729). 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 4727	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 4730 for a more detailed explanation. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 4765 is slightly shorter when the biconditional of ax-ac 4727 is expanded into implication and negation. Standard textbook versions of AC are derived as ac8 4746, ac5 4735, and ac7 4731. The Axiom of Regularity ax-reg 4576 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 4725. Equivalents to AC are the well-ordering theorem weth 4770 and Zorn's lemma zorn 4780. See ac4 4733 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 4728	Axiom of Choice expressed with fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 4727.
|- 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 4729	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 4730 is easier to understand.) Note: aceq0 4713 shows the logical equivalence to ax-ac 4727.
|- 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 4730	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 4727 can be established by chaining aceq0 4713 and aceq2 4714. A standard textbook version of AC is derived from this one in aceq6a 4724, and this version of AC is derived from the textbook version in aceq6b 4725. 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 4587. The key theorem for this (used in the proof of aceq6b 4725) is preleq 4586. 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 4731	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 4732	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 4733	Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 4735, 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 4745.
|- E.fA.z e. x (z =/= (/) -> (f` z) e. z)
Theorem	ac4c 4734	Equivalent of Axiom of Choice (class version)
|- A e. V   =>   |- E.fA.x e. A (x =/= (/) -> (f` x) e. x)
Theorem	ac5 4735	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 4733.
|- A e. V   =>   |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))
Theorem	ac5b 4736	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 4737	Lemma for ac6 4738.
Theorem	ac6 4738	Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a large 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 4739, 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