mmtheorems50 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4901-5000 - Page 50 of 108

Type	Label	Description
Statement
Theorem	alephfplem4 4901	Lemma for alephfp 4902.
Theorem	alephfp 4902	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 4903 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 4903	The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 4902 for an actual example of a fixed point. Compare the inequality alephle 4886 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 4904	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 4905	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 4906	A nonempty set that is a subset of its union is infinite.
|- A e. V   =>   |- ((A =/= (/) /\ A (_ U.A) -> om ~<_ A)
Cofinality
Theorem	cflem 4907	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 4908	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 4909	Cofinality is a function on the class of ordinal numbers.
|- cf Fn On
Theorem	cfub 4910	An upper bound on cofinality.
|- (cf` A) (_ |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A (_ U.y))}
Theorem	cflim 4911	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 4912	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.
|- (cf` (/)) = (/)
Theorem	cardcf 4913	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.
|- (card` (cf` A)) = (cf` A)
Theorem	cflecard 4914	Cofinality is bounded by the cardinality of its argument.
|- (cf` A) (_ (card` A)
Theorem	cfle 4915	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.
|- (cf` A) (_ A
Theorem	cfeq0 4916	Only the ordinal zero has cofinality zero.
|- (A e. On -> ((cf` A) = (/) <-> A = (/)))
Theorem	cfsuc 4917	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102.
|- (A e. On -> (cf` suc A) = 1o)
Theorem	cfom 4918	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 4919	Extend class definition to include cardinal number addition.
class +c
Definition	df-cda 4920	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 4922 for its value and a description.
|- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
Theorem	cdavalt 4921	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	cdaval 4922	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 4833, carddom 4838, and cardsdom 4839. 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 4923	Cardinal addition dominates union.
|- A e. V   &   |- B e. V   =>   |- (A u. B) ~<_ (A +c B)
Theorem	cdaun 4924	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	pm110.643 4925	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 4728), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4574. The comment for cdaval 4922 explains why we use ~~ instead of =.
|- (1o +c 1o) ~~ 2o
Theorem	cdaen 4926	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	cda0en 4927	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 4928	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 4929	One times a cardinal number.
|- A e. V   =>   |- (A X. 1o) ~~ A
Theorem	xp2cda 4930	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
|- A e. V   =>   |- (A X. 2o) = (A +c A)
Theorem	cdacomen 4931	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 4932	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 4933	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 4934	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 4935	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 4936	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 4937	A set is dominated by its cardinal sum with another.
|- A e. V   &   |- B e. V   =>   |- A ~<_ (A +c B)
Theorem	cdafi 4938	The cardinal sum of two finite sets is finite.
|- ((A ~< om /\ B ~< om) -> (A +c B) ~< om)
Theorem	cdainf 4939	A set is infinite iff the cardinal sum with itself is infinite.
|- A e. V   =>   |- (om ~<_ A <-> om ~<_ (A +c A))
ZFC Axioms with no distinct variable requirements
Theorem	nd1 4940	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x y e. z)
Theorem	nd2 4941	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x z e. y)
Theorem	nd3 4942	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z x e. y)
Theorem	nd4 4943	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z y e. x)
Theorem	nd5 4944	A lemma for proving conditionless ZFC axioms.
|- (-. A.y y = x -> (z = y -> A.x z = y))
Theorem	axextnd 4945	A version of the Axiom of Extensionality with no distinct variable conditions.
|- E.x((x e. y <-> x e. z) -> y = z)
Theorem	axrepndlem1 4946	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepndlem2 4947	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepnd 4948	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)))
Theorem	axunndlem1 4949	Lemma for the Axiom of Union with no distinct variable conditions.
Theorem	axunnd 4950	A version of the Axiom of Union with no distinct variable conditions.
|- E.xA.y(E.x(y e. x /\ x e. z) -> y e. x)
Theorem	axpowndlem1 4951	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem2 4952	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem3 4953	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem4 4954	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpownd 4955	A version of the Axiom of Power Sets with no distinct variable conditions.
|- (-. x = y -> E.xA.y(A.x(E.z x e. y -> A.y x e. z) -> y e. x))
Theorem	axregndlem1 4956	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregndlem2 4957	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregnd 4958	A version of the Axiom of Regularity with no distinct variable conditions.
|- (x e. y -> E.x(x e. y /\ A.z(z e. x -> -. z e. y)))
Theorem	axinfndlem1 4959	Lemma for the Axiom of Infinity with no distinct variable conditions.
Theorem	axinfnd 4960	A version of the Axiom of Infinity with no distinct variable conditions.
|- E.x(y e. z -> (y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
Theorem	axacndlem1 4961	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem2 4962	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem3 4963	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem4 4964	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem5 4965	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacnd 4966	A version of the Axiom of Choice with no distinct variable conditions.
|- E.xA.yA.z(A.x(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	zfcndext 4967	Axiom of Extensionality, reproved from conditionless ZFC version and predicate calculus.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	zfcndrep 4968	Axiom of Replacement, reproved from conditionless ZFC axioms.
|- (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))
Theorem	zfcndun 4969	Axiom of Union, reproved from conditionless ZFC axioms.
|- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
Theorem	zfcndpow 4970	Axiom of Power Sets, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 2772.
|- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
Theorem	zfcndreg 4971	Axiom of Regularity, reproved from conditionless ZFC axioms..
|- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)