mmtheorems47 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 105

Type	Label	Description
Statement
Theorem	rankr1a 4601	A relationship between rank and R1, clearly equivalent to ssrankr1 4600 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 4623 for the subset verion. (Contributed by Raph Levien, 29-May-2004.)
|- A e. V   =>   |- (B e. On -> (A e. (R1` B) <-> (rank` A) e. B))
Theorem	r1val2 4602	The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113.
|- (A e. On -> (R1` A) = {x | (rank` x) e. A})
Theorem	r1val3 4603	The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113.
|- (A e. On -> (R1` A) = U_x e. A P~{y | (rank` y) e. x})
Theorem	rankel 4604	The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79.
|- B e. V   =>   |- (A e. B -> (rank` A) e. (rank` B))
Theorem	rankval3 4605	The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79.
|- A e. V   =>   |- (rank` A) = |^|{x e. On | A.y e. A (rank` y) e. x}
Theorem	bndrank 4606	Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80.
|- (E.x e. On A.y e. A (rank` y) (_ x -> A e. V)
Theorem	unbndrank 4607	The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80.
|- (-. A e. V -> A.x e. On E.y e. A x e. (rank` y))
Theorem	rankpw 4608	The rank of a power set. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   =>   |- (rank` P~A) = suc (rank` A)
Theorem	ranklim 4609	The rank of a set belongs to a limit ordinal iff the rank of its power set does.
|- (Lim B -> ((rank` A) e. B <-> (rank` P~A) e. B))
Theorem	r1pw 4610	A stronger property of R1 than rankpw 4608. The latter merely proves that R1 of the successor is a power set, but here we prove that if A is in the cumulative hierarchy, then P~A is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.)
|- (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B)))
Theorem	r1pwcl 4611	The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.)
|- (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B)))
Theorem	rankss 4612	The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80.
|- B e. V   =>   |- (A (_ B -> (rank` A) (_ (rank` B))
Theorem	ranksn 4613	The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112.
|- A e. V   =>   |- (rank` {A}) = suc (rank` A)
Theorem	rankuni2 4614	The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112.
|- A e. V   =>   |- (rank` U.A) = U_x e. A (rank` x)
Theorem	rankun 4615	The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112.
|- A e. V   &   |- B e. V   =>   |- (rank` (A u. B)) = ((rank` A) u. (rank` B))
Theorem	rankpr 4616	The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   &   |- B e. V   =>   |- (rank` {A, B}) = suc ((rank` A) u. (rank` B))
Theorem	rankop 4617	The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107.
|- A e. V   &   |- B e. V   =>   |- (rank` <.A, B>.) = suc suc ((rank` A) u. (rank` B))
Theorem	r1rankid 4618	Any set is a subset of the hierarchy of its rank.
|- (A e. B -> A (_ (R1` (rank` A)))
Theorem	rankonid 4619	The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse.
|- (A e. On <-> (rank` A) = A)
Theorem	rankeq0 4620	A set is empty iff its rank is empty.
|- A e. V   =>   |- (A = (/) <-> (rank` A) = (/))
Theorem	rankr1id 4621	The rank of the hierarchy of an ordinal number is itself.
|- (A e. On <-> (rank` (R1` A)) = A)
Theorem	rankuni 4622	The rank of a union. Part of Exercise 4 of [Kunen] p. 107.
|- (rank` U.A) = U.(rank` A)
Theorem	rankr1b 4623	A relationship between rank and R1. See rankr1a 4601 for the membership version.
|- A e. V   =>   |- (B e. On -> (A (_ (R1` B) <-> (rank` A) (_ B))
Theorem	ranksuc 4624	The rank of a successor.
|- A e. V   =>   |- (rank` suc A) = suc (rank` A)
Theorem	rankuniss 4625	Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   =>   |- (rank` U.A) (_ (rank` A)
Theorem	rankval4 4626	The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204.
|- A e. V   =>   |- (rank` A) = U_x e. A suc (rank` x)
Theorem	rankbnd 4627	The rank of a set is bounded by a bound for the successor of its members.
|- A e. V   =>   |- (A.x e. A suc (rank` x) (_ B <-> (rank` A) (_ B)
Theorem	rankbnd2 4628	The rank of a set is bounded by the successor of a bound for its members.
|- A e. V   =>   |- (B e. On -> (A.x e. A (rank` x) (_ B <-> (rank` A) (_ suc B))
Theorem	rankc1 4629	A relationship that can be used for computation of rank.
|- A e. V   =>   |- (A.x e. A (rank` x) e. (rank` U.A) <-> (rank` A) = (rank` U.A))
Theorem	rankc2 4630	A relationship that can be used for computation of rank.
|- A e. V   =>   |- (E.x e. A (rank` x) = (rank` U.A) -> (rank` A) = suc (rank` U.A))
Theorem	rankelun 4631	Rank membership is inherited by union.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` (A u. B)) e. (rank` (C u. D)))
Theorem	rankelpr 4632	Rank membership is inherited by unordered pairs.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` {A, B}) e. (rank` {C, D}))
Theorem	rankelop 4633	Rank membership is inherited by ordered pairs.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- (((rank` A) e. (rank` C) /\ (rank` B) e. (rank` D)) -> (rank` <.A, B>.) e. (rank` <.C, D>.))
Theorem	rankxpl 4634	A lower bound on the rank of a cross product.
|- A e. V   &   |- B e. V   =>   |- ((A X. B) =/= (/) -> (rank` (A u. B)) (_ (rank` (A X. B)))
Theorem	rankxpu 4635	An upper bound on the rank of a cross product.
|- A e. V   &   |- B e. V   =>   |- (rank` (A X. B)) (_ suc suc (rank` (A u. B))
Theorem	rankxplim 4636	The rank of a cross product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 4639 for the successor case.
|- A e. V   &   |- B e. V   =>   |- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = (rank` (A u. B)))
Theorem	rankxplim2 4637	If the rank of a cross product is a limit ordinal, so is the rank of the union of its arguments.
|- A e. V   &   |- B e. V   =>   |- (Lim (rank` (A X. B)) -> Lim (rank` (A u. B)))
Theorem	rankxplim3 4638	The rank of a cross product is a limit ordinal iff its union is.
|- A e. V   &   |- B e. V   =>   |- (Lim (rank` (A X. B)) <-> Lim U.(rank` (A X. B)))
Theorem	rankxpsuc 4639	The rank of a cross product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 4636 for the limit ordinal case.
|- A e. V   &   |- B e. V   =>   |- (((rank` (A u. B)) = suc C /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = suc suc (rank` (A u. B)))
Scott's trick; collection principle; Hilbert's epsilon
Theorem	scottex 4640	Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set.
|- {x e. A | A.y e. A (rank` x) (_ (rank` y)} e. V
Theorem	scott0 4641	Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. A is empty).
|- (A = (/) <-> {x e. A | A.y e. A (rank` x) (_ (rank` y)} = (/))
Theorem	scottexs 4642	Theorem scheme version of scottex 4640. The collection of all x of minimum rank such that ph(x) is true, is a set.
|- {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} e. V
Theorem	scott0s 4643	Theorem scheme version of scott0 4641. The collection of all x of minimum rank such that ph(x) is true, is not empty iff there is an x such that ph(x) holds.
|- (E.xph <-> {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))} =/= (/))
Theorem	cplem1 4644	Lemma for the Collection Principle cp 4646.
Theorem	cplem2 4645	Lemma for the Collection Principle cp 4646.
Theorem	cp 4646	Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 4640 that collapses a proper class into a set of minimum rank. The wff ph can be thought of as ph(x, y). Scheme "Collection Principle" of [Jech] p. 72.
|- E.wA.x e. z (E.yph -> E.y e. w ph)
Theorem	bnd 4647	A very strong generalization of the Axiom of Replacement (compare zfrep6 3554), derived from the Collection Principle cp 4646. Its strength lies in the rather profound fact that ph(x, y) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom.
|- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)
Theorem	bnd2 4648	A variant of the Boundedness Axiom bnd 4647 that picks a subset z out of a possibly proper class B in which a property is true.
|- A e. V   =>   |- (A.x e. A E.y e. B ph -> E.z(z (_ B /\ A.x e. A E.y e. z ph))
Theorem	kardex 4649	The collection of all sets equinumerous to a set A and having least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222.
|- {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))} e. V
Theorem	karden 4650	If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 4755). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 4649 justify the definition of kard. The restriction to least rank prevents the proper class that would result from {x | x ~~ A}.
|- A e. V   &   |- B e. V   &   |- C = {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (