mmtheorems48 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4701-4800 - Page 48 of 105

Type	Label	Description
Statement
Theorem	kmlem13 4701	Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4.
Theorem	kmlem14 4702	Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4.
Theorem	kmlem15 4703	Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4.
Theorem	kmlem16 4704	Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4.
Theorem	aceqkm 4705	Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and Maes' AC ackm 4706. The proof consists of lemmas kmlem1 4689 through kmlem16 4704 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing aceq5 4664 with pm4.2 170) establishes the AC equivalence shown by Mae's writeup. The left-hand-side AC shown here was chosen because it is shorter to display.
|- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.yA.zE.vA.u((y e. x /\ (z e. y -> ((v e. x /\ -. y = v) /\ z e. v))) \/ (-. y e. x /\ (z e. x -> ((v e. z /\ v e. y) /\ ((u e. z /\ u e. y) -> u = v))))))
Theorem	ackm 4706	A remarkable equivalent to the Axiom of Choice that has only 5 quantifiers (when expanded to e., = primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem aceqkm 4705. Maes found this version of AC in April, 2004 (replacing a longer version, also with 5 quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier (e.,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf). The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html.
|- A.xE.yA.zE.vA.u((y e. x /\ (z e. y -> ((v e. x /\ -. y = v) /\ z e. v))) \/ (-. y e. x /\ (z e. x -> ((v e. z /\ v e. y) /\ ((u e. z /\ u e. y) -> u = v)))))
AC equivalents: well ordering, Zorn's lemma
Theorem	numthlem 4707	Lemma for numth 4708.
Theorem	numth 4708	Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84.
|- A e. V   =>   |- E.x e. On E.f f:x-1-1-onto->A
Theorem	numth2 4709	Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84.
|- E.x e. On x ~~ A
Theorem	numthcor 4710	Any set is strictly dominated by some ordinal.
|- (A e. B -> E.x e. On A ~< x)
Theorem	weth 4711	Well-ordering theorem: any set A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904.
|- A e. V   =>   |- E.x x We A
Theorem	zorn2lem1 4712	Lemma for zorn2 4720.
Theorem	zorn2lem2 4713	Lemma for zorn2 4720.
Theorem	zorn2lem3 4714	Lemma for zorn2 4720.
Theorem	zorn2lem4 4715	Lemma for zorn2 4720.
Theorem	zorn2lem5 4716	Lemma for zorn2 4720.
Theorem	zorn2lem6 4717	Lemma for zorn2 4720.
Theorem	zorn2lem7 4718	Lemma for zorn2 4720.
Theorem	zornlem 4719	Lemma for zorn 4721.
Theorem	zorn2 4720	Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set A (with an ordering relation R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 4712 through zorn2lem7 4718; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 4718.
|- A e. V   =>   |- ((R Po A /\ A.w((w (_ A /\ R Or w) -> E.x e. A A.z e. w (zRx \/ z = x))) -> E.x e. A A.y e. A -. xRy)
Theorem	zorn 4721	Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 4720 for a version with general partial orderings.
|- A e. V   =>   |- (A.z((z (_ A /\ A.x e. z A.y e. z (x (_ y \/ y (_ x)) -> U.z e. A) -> E.x e. A A.y e. A -. x (. y)
Theorem	fodom 4722	An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 4673. AC is not needed for finite sets - see fodomfi 4492.
|- A e. V   =>   |- (F:A-onto->B -> B ~<_ A)
Theorem	fodomg 4723	An onto function implies dominance of domain over range.
|- (A e. C -> (F:A-onto->B -> B ~<_ A))
Theorem	fodomb 4724	Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93.
|- A e. V   =>   |- ((A =/= (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A))
Theorem	brdom3 4725	Equivalence to a dominance relation.
|- A e. V   &   |- B e. V   =>   |- (A ~<_ B <-> E.f(A.xE*y xfy /\ A.x e. A E.y e. B yfx))
Theorem	brdom5 4726	An equivalence to a dominance relation.
|- A e. V   &   |- B e. V   =>   |- (A ~<_ B <-> E.f(A.x e. B E*y xfy /\ A.x e. A E.y e. B yfx))
Theorem	brdom4 4727	An equivalence to a dominance relation.
|- A e. V   &   |- B e. V   =>   |- (A ~<_ B <-> E.f(A.x e. B E*y(y e. A /\ xfy) /\ A.x e. A E.y e. B yfx))
Theorem	brdom7disj 4728	An equivalence to a dominance relation for disjoint sets.
|- A e. V   &   |- B e. V   &   |- (A i^i B) = (/)   =>   |- (A ~<_ B <-> E.f(A.x e. B E*y(y e. A /\ {x, y} e. f) /\ A.x e. A E.y e. B {y, x} e. f))
Theorem	brdom6disj 4729	An equivalence to a dominance relation for disjoint sets.
|- A e. V   &   |- B e. V   &   |- (A i^i B) = (/)   =>   |- (A ~<_ B <-> E.f(A.x e. B E*y{x, y} e. f /\ A.x e. A E.y e. B {y, x} e. f))
Theorem	imadomg 4730	An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92.
|- (A e. B -> (Fun F -> (F"A) ~<_ A))
Theorem	fnrndomg 4731	The range of a function is dominated by its domain.
|- (A e. B -> (F Fn A -> ran F ~<_ A))
Theorem	unidom 4732	An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98.
|- A e. V   &   |- B e. V   =>   |- (A.x e. A x ~<_ B -> U.A ~<_ (A X. B))
Theorem	unidomg 4733	An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98.
|- ((A e. C /\ B e. D /\ A.x e. A x ~<_ B) -> U.A ~<_ (A X. B))
Theorem	uniimadom 4734	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99.
|- A e. V   &   |- B e. V   =>   |- ((Fun F /\ A.x e. A (F` x) ~<_ B) -> U.(F"A) ~<_ (A X. B))
Theorem	uniimadomf 4735	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 4734 uses a bound-variable hypothesis in place of a distinct variable condition.
|- (y e. F -> A.x y e. F)   &   |- A e. V   &   |- B e. V   =>   |- ((Fun F /\ A.x e. A (F` x) ~<_ B) -> U.(F"A) ~<_ (A X. B))
Theorem	iundom 4736	An upper bound for the cardinality of an indexed union. C depends on x and should be thought of as C(x).
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A.x e. A C ~<_ B -> U_x e. A C ~<_ (A X. B))
Cardinal numbers
Syntax	ccrd 4737	Extend class definition to include the cardinal size function.
class card
Syntax	cale 4738	Extend class definition to include the aleph function.
class aleph
Syntax	ccf 4739	Extend class definition to include the cofinality function.
class cf
Definition	df-card 4740	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. See cardval 4750 for its value, cardval2 4778 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 4755. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function.
|- card = {<.x, y>. | y = |^|{z e. On | z ~~ x}}
Definition	df-aleph 4741	Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 4786, alephsuc 4789, and alephlim 4787. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it.
|- aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)
Definition	df-cf 4742	Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 4829 for its value and a description.
|- cf = {<.x, y>. | (x e. On /\ y = |^|{z | E.w(z = (card` w) /\ (w (_ x /\ A.v e. x E.u e. w v (_ u))})}
Theorem	oncardval 4743	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 4750, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
Theorem	oncardon 4744	The cardinal number of an ordinal number is an ordinal number. Unlike cardon 4751, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) e. On)
Theorem	oncardid 4745	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 4752, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) ~~ A)
Theorem	cardonle 4746	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85.
|- (A e. On -> (card` A) (_ A)
Theorem	card0 4747	The cardinality of the empty set is the empty set.
|- (card` (/)) = (/)
Theorem	cardnn 4748	The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90.
|- (A e. om -> (card` A) = A)
Theorem	cardom 4749	The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133.
|- (card` om) = om
Theorem	cardval 4750	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 4778 for a simpler version of its value.
|- (card` A) = |^|{x e. On | x ~~ A}
Theorem	cardon 4751	The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. Unlike Takeuti/Zaring's proposition, we need the Axiom of Choice (in cardval 4750) because of our slightly different definition of of cardinal number.
|- (card` A) e. On
Theorem	cardid 4752	Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85.
|- (card` A) ~~ A
Theorem	oncard 4753	A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85.
|- (E.x A = (card` x) <-> A = (card` A))
Theorem	cardne 4754	No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85.
|- (A e. (card` B) -> -. A ~~ B)
Theorem	carden 4755	Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof. The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank (see karden 4650).
|- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
Theorem	cardeq0 4756	Only the empty set has cardinality zero.
|- (A e. B -> ((card` A) = (/) <-> A = (/)))
Theorem	cardsn 4757	A singleton has cardinality one.
|- (A e. B -> (card` {A}) = 1o)
Theorem	carddomi 4758	Two sets have the dominance relationship if their cardinalities have the subset relationship.
|- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))
Theorem	carddom 4759	Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232.
|- ((A e. C /\ B e. D) -> ((card` A) (_ (card` B) <-> A ~<_ B))
Theorem	cardsdom 4760	Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310.
|- ((A e. C /\ B e. D) -> ((card` A) e. (card` B) <-> A ~< B))
Theorem	domtri 4761	Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice.
|- ((A e. C /\ B e. D) -> (A ~<_ B <-> -. B ~< A))
Theorem	entri 4762	Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242.
|- ((A e. C /\ B e. D) -> (A ~< B \/ A ~~ B \/ B ~< A))
Theorem	entri2 4763	Trichotomy of dominance and strict dominance.
|- ((A e. C /\ B e. D) -> (A ~<_ B \/ B ~<