mmtheorems49 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4801-4900 - Page 49 of 108

Type	Label	Description
Statement
Theorem	brdom3 4801	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 4802	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 4803	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 4804	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 4805	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 4806	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 4807	The range of a function is dominated by its domain.
|- (A e. B -> (F Fn A -> ran F ~<_ A))
Theorem	unidom 4808	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 4809	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 4810	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 4811	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 4810 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 4812	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 4813	Extend class definition to include the cardinal size function.
class card
Syntax	cale 4814	Extend class definition to include the aleph function.
class aleph
Syntax	ccf 4815	Extend class definition to include the cofinality function.
class cf
Definition	df-card 4816	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 4826 for its value, cardval2 4855 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 4831. 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 4817	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 4863, alephsuc 4866, and alephlim 4864. 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 4818	Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 4906 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 4819	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 4826, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
Theorem	oncardon 4820	The cardinal number of an ordinal number is an ordinal number. Unlike cardon 4827, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) e. On)
Theorem	oncardid 4821	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 4828, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) ~~ A)
Theorem	cardonle 4822	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 4823	The cardinality of the empty set is the empty set.
|- (card` (/)) = (/)
Theorem	cardnn 4824	The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90.
|- (A e. om -> (card` A) = A)
Theorem	cardom 4825	The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133.
|- (card` om) = om
Theorem	cardval 4826	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 4855 for a simpler version of its value.
|- (card` A) = |^|{x e. On | x ~~ A}
Theorem	cardon 4827	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 4826) because of our slightly different definition of of cardinal number.
|- (card` A) e. On
Theorem	cardid 4828	Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85.
|- (card` A) ~~ A
Theorem	oncard 4829	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 4830	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 4831	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 4726).
|- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
Theorem	cardeq0 4832	Only the empty set has cardinality zero.
|- (A e. B -> ((card` A) = (/) <-> A = (/)))
Theorem	card1 4833	A set has cardinality one iff it is a singleton.
|- ((card` A) = 1o <-> E.x A = {x})
Theorem	cardsn 4834	A singleton has cardinality one.
|- (A e. B -> (card` {A}) = 1o)
Theorem	carddomi 4835	Two sets have the dominance relationship if their cardinalities have the subset relationship.
|- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))
Theorem	carddom 4836	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 4837	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 4838	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 4839	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 4840	Trichotomy of dominance and strict dominance.
|- ((A e. C /\ B e. D) -> (A ~<_ B \/ B ~< A))
Theorem	entri3 4841	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 4842	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 4523, sucdomi 4524, and finsucdom (future).
|- ((A e. om /\ B e. C) -> (A ~< B <-> suc A ~<_ B))
Theorem	unxpdomlem 4843	Lemma for unxpdom 4844.
Theorem	unxpdom 4844	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 4845	Corollary of unxpdom 4844.
|- A e. V   &   |- B e. V   =>   |- ((1o ~< A /\ B ~<_ A) -> (A u. B) ~<_ (A X. A))
Theorem	sucxpdom 4846	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 4847	Strict dominance implies ordinal membership.
|- ((A e. On /\ B e. On) -> (A ~< B -> A e. B))
Theorem	sdomsdomcard 4848	A set strictly dominates iff its cardinal strictly dominates.
|- (A ~< B <-> A ~< (card` B))
Theorem	cardidm 4849	The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85.
|- (card` (card` A)) = (card` A)
Theorem	canth3 4850	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 4851	An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91.
|- (om (_ (card` A) <-> Lim (card` A))
Theorem	cardsdomel 4852	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93 (use cardsdom 4837 to obtain the exact proposition from this one).
|- (A e. On -> (A ~< B <-> A e. (card` B)))
Theorem	iscard 4853	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 4854	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 4855	An alternate version of the value of the cardinal number of a set. Compare cardval 4826. This theorem could be used to give us a simpler definition of card in place of df-card 4816. It apparently does not occur in the literature.
|- (card` A) = {x e. On | x ~< A}
Theorem	ondomon 4856	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 4857	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 4858	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 4859	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 4860	The smallest ordinal that strictly dominates a set is a cardinal.
|- (A e. B -> (card` |^|{x e. On | A ~< x