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Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 105
TypeLabelDescription
Statement
 
Theoremrankr1a 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))
 
Theoremr1val2 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})
 
Theoremr1val3 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})
 
Theoremrankel 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))
 
Theoremrankval3 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}
 
Theorembndrank 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)
 
Theoremunbndrank 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))
 
Theoremrankpw 4608 The rank of a power set. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   =>   |- (rank` P~A) = suc (rank` A)
 
Theoremranklim 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))
 
Theoremr1pw 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)))
 
Theoremr1pwcl 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)))
 
Theoremrankss 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))
 
Theoremranksn 4613 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112.
|- A e. V   =>   |- (rank` {A}) = suc (rank` A)
 
Theoremrankuni2 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)
 
Theoremrankun 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))
 
Theoremrankpr 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))
 
Theoremrankop 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))
 
Theoremr1rankid 4618 Any set is a subset of the hierarchy of its rank.
|- (A e. B -> A (_ (R1`
 (rank` A)))
 
Theoremrankonid 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)
 
Theoremrankeq0 4620 A set is empty iff its rank is empty.
|- A e. V   =>   |- (A = (/) <-> (rank` A) = (/))
 
Theoremrankr1id 4621 The rank of the hierarchy of an ordinal number is itself.
|- (A e. On <-> (rank` (R1`
 A)) = A)
 
Theoremrankuni 4622 The rank of a union. Part of Exercise 4 of [Kunen] p. 107.
|- (rank` U.A) = U.(rank` A)
 
Theoremrankr1b 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))
 
Theoremranksuc 4624 The rank of a successor.
|- A e. V   =>   |- (rank` suc A) = suc (rank` A)
 
Theoremrankuniss 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)
 
Theoremrankval4 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)
 
Theoremrankbnd 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)
 
Theoremrankbnd2 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))
 
Theoremrankc1 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))
 
Theoremrankc2 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))
 
Theoremrankelun 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)))
 
Theoremrankelpr 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}))
 
Theoremrankelop 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>.))
 
Theoremrankxpl 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)))
 
Theoremrankxpu 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))
 
Theoremrankxplim 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)))
 
Theoremrankxplim2 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)))
 
Theoremrankxplim3 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)))
 
Theoremrankxpsuc 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
 
Theoremscottex 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
 
Theoremscott0 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)} = (/))
 
Theoremscottexs 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
 
Theoremscott0s 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)))} =/= (/))
 
Theoremcplem1 4644 Lemma for the Collection Principle cp 4646.
 
Theoremcplem2 4645 Lemma for the Collection Principle cp 4646.
 
Theoremcp 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)
 
Theorembnd 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)
 
Theorembnd2 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))
 
Theoremkardex 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
 
Theoremkarden 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) (_ (