Metamath Proof Explorer - mmtheorems36

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Statement List for Metamath Proof Explorer - 3501-3600 - Page 36 of 58

Type	Label	Description
Statement
Theorem	r1ord3 3501	Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478.
|- ((A e. On /\ B e. On) -> (A (_ B -> (R1` A) (_ (R1` B)))
Theorem	r1val1 3502	The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202.
|- (A e. On -> (R1` A) = U.x e. A P~(R1` x))
Theorem	tz9.12lem1 3503	Lemma for tz9.12 3506.
|- A e. V   &   |- F = {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}}   =>   |- (F"A) (_ On
Theorem	tz9.12lem2 3504	Lemma for tz9.12 3506.
|- A e. V   &   |- F = {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}}   =>   |- suc U.(F"A) e. On
Theorem	tz9.12lem3 3505	Lemma for tz9.12 3506.
|- A e. V   &   |- F = {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}}   =>   |- (A.x e. A E.y e. On x e. (R1` y) -> A e. (R1` suc suc U.(F"A)))
Theorem	tz9.12 3506	A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 3503 through tz9.12lem3 3505.
|- A e. V   =>   |- (A.x e. A E.y e. On x e. (R1` y) -> E.y e. On A e. (R1` y))
Theorem	tz9.13 3507	Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78.
|- A e. V   =>   |- E.x e. On A e. (R1` x)
Theorem	tz9.13g 3508	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 3507 expresses the class existence requirement as an antecedent.
|- (A e. B -> E.x e. On A e. (R1` x))
Theorem	rankwflem 3509	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 3508 is useful in proofs of theorems about the rank function.
|- (A e. B -> E.x e. On A e. (R1` suc x))
Theorem	jech9.3 3510	Every set belongs to some value of the cumulative hierarchy of sets function R1, i.e. the indexed union of all values of R1 is the universe. Lemma 9.3 of [Jech] p. 71.
|- U.x e. On (R1` x) = V
Theorem	unir1 3511	The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281.
|- U.(R1"On) = V
Theorem	rankval 3512	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition).
|- A e. V   =>   |- (rank` A) = |^|{x e. On | A e. (R1` suc x)}
Theorem	rankvalg 3513	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This more general variant of rankval 3512 expresses the class existence requirement as an antecedent instead of a hypothesis.
|- (A e. B -> (rank` A) = |^|{x e. On | A e. (R1` suc x)})
Theorem	rankval2 3514	Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478.
|- (A e. B -> (rank` A) = |^|{x e. On | A (_ (R1` x)})
Theorem	rankon 3515	The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79.
|- (rank` A) e. On
Theorem	rankid 3516	Identity law for the rank function.
|- A e. V   =>   |- A e. (R1` suc (rank` A))
Theorem	rankr1lem 3517	Lemma for rankr1 3518.
|- A e. V   =>   |- (B e. On -> (-. A e. (R1` B) -> B (_ (rank` A)))
Theorem	rankr1 3518	A relationship between the rank function and the cumulative hierarchy of sets function R1. Proposition 9.15(2) of [TakeutiZaring] p. 79.
|- A e. V   =>   |- (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B)))
Theorem	rankr1g 3519	A relationship between the rank function and the cumulative hierarchy of sets function R1. Proposition 9.15(2) of [TakeutiZaring] p. 79. This more general variant of rankr1 3518 expresses the class existence requirement as an antecedent.
|- (A e. C -> (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B))))
Theorem	ssrankr1 3520	A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets R1. Proposition 9.15(3) of [TakeutiZaring] p. 79.
|- A e. V   =>   |- (B e. On -> (B (_ (rank` A) <-> -. A e. (R1` B)))
Theorem	rankr1a 3521	A relationship between rank and R1, clearly equivalent to ssrankr1 3520 and friends through trichotomy, but in Raph's opinion considerably more intuitive. (Contributed by Raph Levien, 29-May-04.)
|- A e. V   =>   |- (B e. On -> (A e. (R1` B) <-> (rank` A) e. B))
Theorem	r1val2 3522	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 3523	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 3524	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 3525	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 3526	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 3527	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 3528	The rank of a power set. Part of Exercise 30 of [Enderton] p. 207.
|- A e. V   =>   |- (rank` P~A) = suc (rank` A)
Theorem	r1pw 3529	A stronger property of R1 than rankpw 3528. The latter merely proves that R1 of the successor is a powerset, 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-04.)
|- (B e. On -> (A e. (R1` B) <-> P~A e. (R1` suc B)))
Theorem	r1pwcl 3530	The cumulative hierarchy of a limit ordinal is closed under powerset. (Contributed by Raph Levien, 29-May-04.)
|- (Lim B -> (A e. (R1` B) <-> P~A e. (R1` B)))
Theorem	rankss 3531	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 3532	The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112.
|- A e. V   =>   |- (rank` {A}) = suc (rank` A)
Theorem	rankuni 3533	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	rankuniss 3534	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	rankun 3535	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 3536	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	r1rankid 3537	Any set is a subset of the hierarchy of its rank.
|- (A e. B -> A (_ (R1` (rank` A)))
Theorem	rankonid 3538	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	rankr1id 3539	The rank of the hierarchy of an ordinal number is itself.
|- (A e. On <-> (rank` (R1` A)) = A)
Theorem	ranklon 3540	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	scottex 3541	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 3542	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 3543	Theorem scheme version of scottex 3541. 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 3544	Theorem scheme version of scott0 3542. 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 3545	Lemma for the Collection Principle cp 3547.
|- C = {y e. B | A.z e. B (rank` y) (_ (rank` z)}   &   |- D = U.x e. A C   =>   |- A.x e. A (-. B = (/) -> -. (B i^i D) = (/))
Theorem	cplem2 3546	Lemma for the Collection Principle cp 3547.
|- A e. V   =>   |- E.yA.x e. A (-. B = (/) -> -. (B i^i y) = (/))
Theorem	cp 3547	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 3541 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 3548	A very strong generalization of the Axiom of Replacement (compare zfrep6 2744), derived from the Collection Principle cp 3547. 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 3549	A variant of the Boundedness Axiom bnd 3548 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 3550	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.
|- A e. V   =>   |- {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))} e. V
Theorem	karden 3551	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 3638). 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 3550 justify the definition of kard.
|- A e. V   &   |- B e. V   &   |- C = {x | (x ~~ A /\ A.y(y ~~ A -> (rank` x) (_ (rank` y)))}   &   |- D = {x | (x ~~ B /\ A.y(y ~~ B -> (rank` x) (_ (rank` y)))}   =>   |- (C = D <-> A ~~ B)
Theorem	aceq1 3552	Equivalence of two versions of the Axiom of Choice ax-ac 1080. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables.
|- (E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u) <-> E.yA.zA.w((z e. w /\ w e. x) -> E.xA.z(E.x((z e. w /\ w e. x) /\ (z e. x /\ x e. y)) <-> z = x)))
Theorem	aceq0 3553	Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 1080.
|- (E.yA.z e. x A.w e. z E!v e. z E.u e. y (z e. u /\ v e. u) <-> E.yA.