mmtheorems49 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4801-4900 - Page 49 of 107

Type	Label	Description
Statement
Theorem	uniimadom 4801	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 4802	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 4801 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 4803	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 4804	Extend class definition to include the cardinal size function.
class card
Syntax	cale 4805	Extend class definition to include the aleph function.
class aleph
Syntax	ccf 4806	Extend class definition to include the cofinality function.
class cf
Definition	df-card 4807	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 4817 for its value, cardval2 4846 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 4822. 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 4808	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 4854, alephsuc 4857, and alephlim 4855. 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 4809	Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 4897 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 4810	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 4817, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
Theorem	oncardon 4811	The cardinal number of an ordinal number is an ordinal number. Unlike cardon 4818, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) e. On)
Theorem	oncardid 4812	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 4819, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) ~~ A)
Theorem	cardonle 4813	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 4814	The cardinality of the empty set is the empty set.
|- (card` (/)) = (/)
Theorem	cardnn 4815	The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90.
|- (A e. om -> (card` A) = A)
Theorem	cardom 4816	The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133.
|- (card` om) = om
Theorem	cardval 4817	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 4846 for a simpler version of its value.
|- (card` A) = |^|{x e. On | x ~~ A}
Theorem	cardon 4818	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 4817) because of our slightly different definition of of cardinal number.
|- (card` A) e. On
Theorem	cardid 4819	Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85.
|- (card` A) ~~ A
Theorem	oncard 4820	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 4821	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 4822	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 4717).
|- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
Theorem	cardeq0 4823	Only the empty set has cardinality zero.
|- (A e. B -> ((card` A) = (/) <-> A = (/)))
Theorem	card1 4824	A set has cardinality one iff it is a singleton.
|- ((card` A) = 1o <-> E.x A = {x})
Theorem	cardsn 4825	A singleton has cardinality one.
|- (A e. B -> (card` {A}) = 1o)
Theorem	carddomi 4826	Two sets have the dominance relationship if their cardinalities have the subset relationship.
|- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))
Theorem	carddom 4827	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 4828	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 4829	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 4830	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 4831	Trichotomy of dominance and strict dominance.
|- ((A e. C /\ B e. D) -> (A ~<_ B \/ B ~< A))
Theorem	entri3 4832	Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275.
|- ((A e. C /\ B e. D) -> (A ~<_ B \/ B ~<_ A))
Theorem	sucdom 4833	Strict dominance of a set over a natural number is the same as dominance over its successor. The proof uses AC and Infinity. It is unclear if a proof without using these is possible, unlike the weaker versions omsucdom 4519, sucdomi 4520, and finsucdom 4523.
|- ((A e. om /\ B e. C) -> (A ~< B <-> suc A ~<_ B))
Theorem	unxpdomlem 4834	Lemma for unxpdom 4835.
Theorem	unxpdom 4835	Cross product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93.
|- ((1o ~< A /\ 1o ~< B) -> (A u. B) ~<_ (A X. B))
Theorem	unxpdom2 4836	Corollary of unxpdom 4835.
|- A e. V   &   |- B e. V   =>   |- ((1o ~< A /\ B ~<_ A) -> (A u. B) ~<_ (A X. A))
Theorem	sucxpdom 4837	Cross product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals, with a proof using AC).
|- (1o ~< A -> suc A ~<_ (A X. A))
Theorem	sdomel 4838	Strict dominance implies ordinal membership.
|- ((A e. On /\ B e. On) -> (A ~< B -> A e. B))
Theorem	sdomsdomcard 4839	A set strictly dominates iff its cardinal strictly dominates.
|- (A ~< B <-> A ~< (card` B))
Theorem	cardidm 4840	The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85.
|- (card` (card` A)) = (card` A)
Theorem	canth3 4841	Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133.
|- (A e. B -> (card` A) e. (card` P~A))
Theorem	cardlim 4842	An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91.
|- (om (_ (card` A) <-> Lim (card` A))
Theorem	cardsdomel 4843	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93 (use cardsdom 4828 to obtain the exact proposition from this one).
|- (A e. On -> (A ~< B <-> A e. (card` B)))
Theorem	iscard 4844	Two ways to express the property of being a cardinal number.
|- ((card` A) = A <-> (A e. On /\ A.x e. A x ~< A))
Theorem	iscard2 4845	Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225.
|- ((card` A) = A <-> (A e. On /\ A.x e. On (A ~~ x -> A (_ x)))
Theorem	cardval2 4846	An alternate version of the value of the cardinal number of a set. Compare cardval 4817. This theorem could be used to give us a simpler definition of card in place of df-card 4807. It apparently does not occur in the literature.
|- (card` A) = {x e. On | x ~< A}
Theorem	ondomon 4847	The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227.
|- (A e. B -> {x e. On | x ~<_ A} e. On)
Theorem	ondomcard 4848	The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228.
|- (A e. B -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})
Theorem	carduni 4849	The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133.
|- (A e. B -> (A.x e. A (card` x) = x -> (card` U.A) = U.A))
Theorem	cardiun 4850	The indexed union of a set of cardinals is a cardinal.
|- (A e. C -> (A.x e. A (card` B) = B -> (card` U_x e. A B) = U_x e. A B))
Theorem	cardmin 4851	The smallest ordinal that strictly dominates a set is a cardinal.
|- (A e. B -> (card` |^|{x e. On | A ~< x}) = |^|{x e. On | A ~< x})
Theorem	cardprc 4852	The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310.
|- -. {x | (card` x) = x} e. V
Theorem	alephfnon 4853	The aleph function is a function on the class of ordinal numbers.
|- aleph Fn On
Theorem	aleph0 4854	The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers om (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written aleph0. Exercise 3 of [TakeutiZaring] p. 91. Also Definition 12(i) of [Suppes] p. 228. From Kuratowski and Mostowski, Set Theory, p. 95: "Aleph...the first letter in the Hebrew alphabet...is also the first letter of the Hebrew word...(einsoph, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism."
|- (aleph` (/)) = om
Theorem	alephlim 4855	Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91.
|- ((A e. B /\ Lim A) -> (aleph` A) = U_x e. A (aleph` x))
Theorem	alephon 4856	An aleph is an ordinal number.
|- (aleph` A) e. On
Theorem	alephsuc 4857	Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91.
|- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
Theorem	alephcard 4858	Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229.
|- (card` (aleph` A)) = (aleph` A)
Theorem	alephnbtwn 4859	No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229.
|- ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
Theorem	alephnbtwn2 4860	No set has equinumerosity between an aleph and its successor aleph.
|- -. ((aleph` A) ~< B /\ B ~< (aleph` suc A))
Theorem	alephsucpw 4861	The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous.)
|- (aleph` suc A) ~<_ P~(aleph` A)
Theorem	aleph1 4862	The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.)
|- (aleph` 1o) ~<_ (2o ^m (aleph` (/)))
Theorem	alephordlem1 4863	Lemma for alephordi 4865.
Theorem	alephordlem2 4864	Lemma for alephordi 4865.
Theorem	alephordi 4865	Strict ordering property of the aleph function.
|- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))
Theorem	alephord 4866	Ordering property of the aleph function.
|- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))
Theorem	alephord2 4867	Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse.
|- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) e. (aleph` B)))
Theorem	alephord2i 4868	Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229.