mmtheorems51 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5001-5100 - Page 51 of 123

Type	Label	Description
Statement
Theorem	cardlim 5001	An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91.
|- (om (_ (card` A) <-> Lim (card` A))
Theorem	cardsdomel 5002	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93 (use cardsdom 4986 to obtain the exact proposition from this one).
|- (A e. On -> (A ~< B <-> A e. (card` B)))
Theorem	iscard 5003	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 5004	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 5005	An alternate version of the value of the cardinal number of a set. Compare cardval 4973. This theorem could be used to give us a simpler definition of card in place of df-card 4962. It apparently does not occur in the literature.
|- (card` A) = {x e. On | x ~< A}
Theorem	ondomon 5006	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 5007	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 5008	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 5009	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 5010	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 5011	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 5012	The aleph function is a function on the class of ordinal numbers.
|- aleph Fn On
Theorem	aleph0 5013	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 5014	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 5015	An aleph is an ordinal number.
|- (aleph` A) e. On
Theorem	alephsuc 5016	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 5017	Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229.
|- (card` (aleph` A)) = (aleph` A)
Theorem	alephnbtwn 5018	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 5019	No set has equinumerosity between an aleph and its successor aleph.
|- -. ((aleph` A) ~< B /\ B ~< (aleph` suc A))
Theorem	alephsucpw 5020	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 5021	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 5022	Lemma for alephordi 5024.
Theorem	alephordlem2 5023	Lemma for alephordi 5024.
Theorem	alephordi 5024	Strict ordering property of the aleph function.
|- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))
Theorem	alephord 5025	Ordering property of the aleph function.
|- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))
Theorem	alephord2 5026	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 5027	Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229.
|- (B e. On -> (A e. B -> (aleph` A) e. (aleph` B)))
Theorem	alephord3 5028	Ordering property of the aleph function.
|- ((A e. On /\ B e. On) -> (A (_ B <-> (aleph` A) (_ (aleph` B)))
Theorem	aleph11 5029	The aleph function is one-to-one.
|- ((A e. On /\ B e. On) -> ((aleph` A) = (aleph` B) <-> A = B))
Theorem	alephsucdom 5030	A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa.
|- (B e. On -> (A ~<_ (aleph` B) <-> A ~< (aleph` suc B)))
Theorem	alephsuc2 5031	An alternate representation of a successor aleph. Using this theorem we could define the aleph function with {z e. On | z ~<_ x} in place of |^|{z e. On | x ~< z} in df-aleph 4963.
|- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})
Theorem	alephgeom 5032	Every aleph is greater than or equal to the set of natural numbers.
|- (A e. On <-> om (_ (aleph` A))
Theorem	alephislim 5033	Every aleph is a limit ordinal.
|- (A e. On <-> Lim (aleph` A))
Theorem	alephle 5034	The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 5051, we will that equality can sometimes hold.)
|- (A e. On -> A (_ (aleph` A))
Theorem	cardaleph 5035	Given any transfinite cardinal number A, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly.
|- ((om (_ A /\ (card` A) = A) -> A = (aleph` |^|{x e. On | A (_ (aleph` x)}))
Theorem	cardalephex 5036	Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse.
|- (om (_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))
Theorem	isinfcard 5037	Two ways to express the property of being a transfinite cardinal.
|- ((om (_ A /\ (card` A) = A) <-> A e. ran aleph)
Theorem	iscard3 5038	Two ways to express the property of being a cardinal number.
|- ((card` A) = A <-> A e. (om u. ran aleph))
Theorem	cardnum 5039	Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in om plus the transfinite alephs.
|- {x | (card` x) = x} = (om u. ran aleph)
Theorem	carduniima 5040	The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104.
|- (A e. B -> (F:A-->(om u. ran aleph) -> U.(F"A) e. (om u. ran aleph)))
Theorem	cardinfima 5041	If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104.
|- (A e. B -> ((F:A-->(om u. ran aleph) /\ E.x e. A (F` x) e. ran aleph) -> U.(F"A) e. ran aleph))
Theorem	alephiso 5042	Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90.
|- aleph Isom E, E (On, {x | (om (_ x /\ (card` x) = x)})
Theorem	alephprc 5043	The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90.
|- -. ran aleph e. V
Theorem	alephsson 5044	The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers.
|- ran aleph (_ On
Theorem	unialeph 5045	The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers.
|- U.ran aleph = On
Theorem	alephfplem1 5046	Lemma for alephfp 5050.
Theorem	alephfplem2 5047	Lemma for alephfp 5050.
Theorem	alephfplem3 5048	Lemma for alephfp 5050.
Theorem	alephfplem4 5049	Lemma for alephfp 5050.
Theorem	alephfp 5050	The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 5051 for an abbreviated version just showing existence.
|- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)   =>   |- (aleph` U.(H"om)) = U.(H"om)
Theorem	alephfp2 5051	The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 5050 for an actual example of a fixed point. Compare the inequality alephle 5034 that holds in general. Note that if x is a fixed point, then aleph` aleph` aleph` ... aleph` x = x.
|- E.x e. On (aleph` x) = x
Theorem	alephval2 5052	An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229.
|- ((A e. On /\ (/) e. A) -> (aleph` A) = |^|{x e. On | A.y e. A (aleph` y) ~< x})
Theorem	alephval3 5053	An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 .
|- (A e. On -> (aleph` A) = |^|{x | ((card` x) = x /\ om (_ x /\ A.y e. A -. x = (aleph` y))})
Theorem	dominf 5054	A nonempty set that is a subset of its union is infinite.
|- A e. V   =>   |- ((A =/= (/) /\ A (_ U.A) -> om ~<_ A)
Cofinality
Theorem	cflem 5055	A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set A.
|- (A e. B -> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
Theorem	cfval 5056	Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A, there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is.
|- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
Theorem	cffnon 5057	Cofinality is a function on the class of ordinal numbers.
|- cf Fn On
Theorem	cfub 5058	An upper bound on cofinality.
|- (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}
Theorem	cflim 5059	Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257.
|- ((A e. B /\ Lim A) -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A = U.y))})
Theorem	cf0 5060	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.
|- (cf` (/)) = (/)
Theorem	cardcf 5061	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.
|- (card` (cf` A)) = (cf` A)
Theorem	cflecard 5062	Cofinality is bounded by the cardinality of its argument.
|- (cf` A) (_ (card` A)
Theorem	cfle 5063	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.
|- (cf` A) (_ A
Theorem	cfeq0 5064	Only the ordinal zero has cofinality zero.
|- (A e. On -> ((cf` A) = (/) <-> A = (/)))
Theorem	cfsuc 5065	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102.
|- (A e. On -> (cf` suc A) = 1o)
Theorem	cfom 5066	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102.
|- (cf` om) = om
Cardinal number arithmetic
Syntax	ccda 5067	Extend class definition to include cardinal number addition.
class +c
Definition	df-cda 5068	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdavali 5070 for its value and a description.
|- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
Theorem	cdaval 5069	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.
|- ((A e. C /\ B e. D) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
Theorem	cdavali 5070	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 4979, carddom 4985, and cardsdom 4986. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.
|- A e. V   &   |- B e. V   =>   |- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))
Theorem	uncdadom 5071	Cardinal addition dominates union.
|- A e. V   &   |- B e. V   =>   |- (A u. B) ~<_ (A +c B)
Theorem	cdaun 5072	Cardinal addition is equinumerous to union for disjoint sets.
|- A e. V   &   |- B e. V   =>   |- ((A i^i B) = (/) -> (A +c B) ~~ (A u. B))
Theorem	cdaung 5073	Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Paul Chapman, 5-Jun-2009.)
|- ((A e. C /\ B e. D /\ (A i^i B) = (/)) -> (A +c B) ~~ (A u. B))
Theorem	pm110.643 5074	1+1=2 for cardinal number addition. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Unlike us, Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 4872), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4715. The comment for cdavali 5070 explains why we use ~~ instead of =.
|- (1o +c 1o) ~~ 2o
Theorem	cdaen 5075	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))
Theorem	cdaeng 5076	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by Paul Chapman, 5-Jun-2009.)
|- (((A e. W /\ B e. X) /\ (C e. Y /\ D e. Z) /\ (A ~~ B /\ C ~~ D)) -> (A +c C) ~~ (B +c D))
Theorem	cda0en 5077	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143.
|- A e. V   =>   |- (A +c (/)) ~~ A
Theorem	cda1en 5078	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.
|- A e. V   =>   |- (A +c 1o) ~~ suc (card` A)
Theorem	xp1en 5079	One times a cardinal number.
|- A e. V   =>   |- (A X. 1o) ~~ A
Theorem	xp2cda 5080	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
|- A e. V   =>   |- (A X. 2o) = (A +c A)
Theorem	cdacomen 5081	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   =>   |- (A +c B) ~~ (B +c A)
Theorem	cdaassen 5082	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A +c B) +c C) ~~ (A +c (B +c C))
Theorem	xpcdaen 5083	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))
Theorem	mapcdaen 5084	Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ^m (B +c C)) ~~ ((A ^m B) X. (A ^m C))
Theorem	cdadom1 5085	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (A +c C) ~<_ (B +c C))
Theorem	cdadom2 5086	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (C +c A) ~<_ (C +c B))
Theorem	cdadom3 5087	A set is dominated by its cardinal sum with another.
|- A e. V   &   |- B e. V   =>   |- A ~<_ (A +c B)
Theorem	cdafi 5088	The cardinal sum of two finite sets is finite.
|- ((A ~< om /\ B ~< om) -> (A +c B) ~< om)
Theorem	cdainf 5089	A set is infinite iff the cardinal sum with itself is infinite.
|- A e. V   =>   |- (om ~<_ A <-> om ~<_ (A +c A))
Theorem	nnacda 5090	The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ((A e. om /\ B e. om) -> (card` (A +c B)) = (A +o B))
Theorem	nnaun 5091	The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.)
|- ((A e. Fin /\ B e. Fin /\ (A i^i B) = (/)) -> (card` (A u. B)) = ((card` A) +o (card` B)))
ZFC Axioms with no distinct variable requirements
Theorem	nd1 5092	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x y e. z)
Theorem	nd2 5093	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x z e. y)
Theorem	nd3 5094	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z x e. y)
Theorem	nd4 5095	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z y e. x)
Theorem	nd5 5096	A lemma for proving conditionless ZFC axioms.
|- (-. A.y y = x -> (z = y -> A.x z = y))
Theorem	axextnd 5097	A version of the Axiom of Extensionality with no distinct variable conditions.
|- E.x((x e. y <-> x e. z) -> y = z)
Theorem	axrepndlem1 5098	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepndlem2 5099	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepnd 5100	A version of the Axiom of Replacement with no distinct variable conditions.
|- E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))