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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | inf3lem5 9501* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9504 for detailed description. (Contributed by NM, 29-Oct-1996.) |
β’ πΊ = (π¦ β V β¦ {π€ β π₯ β£ (π€ β© π₯) β π¦}) & β’ πΉ = (rec(πΊ, β ) βΎ Ο) & β’ π΄ β V & β’ π΅ β V β β’ ((π₯ β β β§ π₯ β βͺ π₯) β ((π΄ β Ο β§ π΅ β π΄) β (πΉβπ΅) β (πΉβπ΄))) | ||
Theorem | inf3lem6 9502* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9504 for detailed description. (Contributed by NM, 29-Oct-1996.) |
β’ πΊ = (π¦ β V β¦ {π€ β π₯ β£ (π€ β© π₯) β π¦}) & β’ πΉ = (rec(πΊ, β ) βΎ Ο) & β’ π΄ β V & β’ π΅ β V β β’ ((π₯ β β β§ π₯ β βͺ π₯) β πΉ:Οβ1-1βπ« π₯) | ||
Theorem | inf3lem7 9503* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9504 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7879. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.) |
β’ πΊ = (π¦ β V β¦ {π€ β π₯ β£ (π€ β© π₯) β π¦}) & β’ πΉ = (rec(πΊ, β ) βΎ Ο) & β’ π΄ β V & β’ π΅ β V β β’ ((π₯ β β β§ π₯ β βͺ π₯) β Ο β V) | ||
Theorem | inf3 9504 |
Our Axiom of Infinity ax-inf 9507 implies the standard Axiom of Infinity.
The hypothesis is a variant of our Axiom of Infinity provided by
inf2 9492, and the conclusion is the version of the Axiom of Infinity
shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are
proved later as axinf2 9509 and zfinf2 9511.) The main proof is provided by
inf3lema 9493 through inf3lem7 9503, and this final piece eliminates the
auxiliary hypothesis of inf3lem7 9503. This proof is due to
Ian Sutherland, Richard Heck, and Norman Megill and was posted
on Usenet as shown below. Although the result is not new, the authors
were unable to find a published proof.
(As posted to sci.logic on 30-Oct-1996, with annotations added.) Theorem: The statement "There exists a nonempty set that is a subset of its union" implies the Axiom of Infinity. Proof: Let X be a nonempty set which is a subset of its union; the latter property is equivalent to saying that for any y in X, there exists a z in X such that y is in z. Define by finite recursion a function F:omega-->(power X) such that F_0 = 0 (See inf3lemb 9494.) F_n+1 = {y<X | y^X subset F_n} (See inf3lemc 9495.) Note: ^ means intersect, < means \in ("element of"). (Finite recursion as typically done requires the existence of omega; to avoid this we can just use transfinite recursion restricted to omega. F is a class-term that is not necessarily a set at this point.) Lemma 1. F_n subset F_n+1. (See inf3lem1 9497.) Proof: By induction: F_0 subset F_1. If y < F_n+1, then y^X subset F_n, so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2. Lemma 2. F_n =/= X. (See inf3lem2 9498.) Proof: By induction: F_0 =/= X because X is not empty. Assume F_n =/= X. Then there is a y in X that is not in F_n. By definition of X, there is a z in X that contains y. Suppose F_n+1 = X. Then z is in F_n+1, and z^X contains y, so z^X is not a subset of F_n, contrary to the definition of F_n+1. Lemma 3. F_n =/= F_n+1. (See inf3lem3 9499.) Proof: Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have F_n+1 = {y<X | y^(X-F_n) = 0}. Let q = {y<X-F_n | y^(X-F_n) = 0}. Then q subset F_n+1. Since X-F_n is not empty by Lemma 2 and q is the set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q and therefore F_n+1 have an element not in F_n. Lemma 4. F_n proper_subset F_n+1. (See inf3lem4 9500.) Proof: Lemmas 1 and 3. Lemma 5. F_m proper_subset F_n, m < n. (See inf3lem5 9501.) Proof: Fix m and use induction on n > m. Basis: F_m proper_subset F_m+1 by Lemma 4. Induction: Assume F_m proper_subset F_n. Then since F_n proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper subset. By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1. (See inf3lem6 9502.) Thus, the inverse of F is a function with range omega and domain a subset of power X, so omega exists by Replacement. (See inf3lem7 9503.) Q.E.D.(Contributed by NM, 29-Oct-1996.) |
β’ βπ₯(π₯ β β β§ π₯ β βͺ π₯) β β’ Ο β V | ||
Theorem | infeq5i 9505 | Half of infeq5 9506. (Contributed by Mario Carneiro, 16-Nov-2014.) |
β’ (Ο β V β βπ₯ π₯ β βͺ π₯) | ||
Theorem | infeq5 9506 | The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 9512.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.) |
β’ (βπ₯ π₯ β βͺ π₯ β Ο β V) | ||
Axiom | ax-inf 9507* |
Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom
is the gateway to "Cantor's paradise" (an expression coined by
Hilbert).
It asserts that given a starting set π₯, an infinite set π¦ built
from it exists. Although our version is apparently not given in the
literature, it is similar to, but slightly shorter than, the Axiom of
Infinity in [FreydScedrov] p. 283
(see inf1 9491 and inf2 9492). More
standard versions, which essentially state that there exists a set
containing all the natural numbers, are shown as zfinf2 9511 and omex 9512 and
are based on the (nontrivial) proof of inf3 9504.
This version has the
advantage that when expanded to primitives, it has fewer symbols than
the standard version ax-inf2 9510. Theorem inf0 9490
shows the reverse
derivation of our axiom from a standard one. Theorem inf5 9514
shows a
very short way to state this axiom.
The standard version of Infinity ax-inf2 9510 requires this axiom along with Regularity ax-reg 9461 for its derivation (as Theorem axinf2 9509 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 9510 instead of this one. The derivation of this axiom from ax-inf2 9510 is shown by Theorem axinf 9513. Proofs should normally use the standard version ax-inf2 9510 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.) |
β’ βπ¦(π₯ β π¦ β§ βπ§(π§ β π¦ β βπ€(π§ β π€ β§ π€ β π¦))) | ||
Theorem | zfinf 9508* | Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
β’ βπ₯(π¦ β π₯ β§ βπ¦(π¦ β π₯ β βπ§(π¦ β π§ β§ π§ β π₯))) | ||
Theorem | axinf2 9509* |
A standard version of Axiom of Infinity, expanded to primitives, derived
from our version of Infinity ax-inf 9507 and Regularity ax-reg 9461.
This theorem should not be referenced in any proof. Instead, use ax-inf2 9510 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.) |
β’ βπ₯(βπ¦(π¦ β π₯ β§ βπ§ Β¬ π§ β π¦) β§ βπ¦(π¦ β π₯ β βπ§(π§ β π₯ β§ βπ€(π€ β π§ β (π€ β π¦ β¨ π€ = π¦))))) | ||
Axiom | ax-inf2 9510* | A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 9511 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9509 above, using our version of Infinity ax-inf 9507 and the Axiom of Regularity ax-reg 9461. We will reference ax-inf2 9510 instead of axinf2 9509 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9507 from ax-inf2 9510 is shown by Theorem axinf 9513. (Contributed by NM, 3-Nov-1996.) |
β’ βπ₯(βπ¦(π¦ β π₯ β§ βπ§ Β¬ π§ β π¦) β§ βπ¦(π¦ β π₯ β βπ§(π§ β π₯ β§ βπ€(π€ β π§ β (π€ β π¦ β¨ π€ = π¦))))) | ||
Theorem | zfinf2 9511* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 9510 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.) |
β’ βπ₯(β β π₯ β§ βπ¦ β π₯ suc π¦ β π₯) | ||
Theorem | omex 9512 |
The existence of omega (the class of natural numbers). Axiom 7 of
[TakeutiZaring] p. 43. This
theorem is proved assuming the Axiom of
Infinity and in fact is equivalent to it, as shown by the reverse
derivation inf0 9490.
A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial Β¬ Ο β V; this would lead to Ο = On by omon 7804 and Fin = V (the universe of all sets) by fineqv 9140. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 7815 through peano5 7820 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.) |
β’ Ο β V | ||
Theorem | axinf 9513* | The first version of the Axiom of Infinity ax-inf 9507 proved from the second version ax-inf2 9510. Note that we didn't use ax-reg 9461, unlike the other direction axinf2 9509. (Contributed by NM, 24-Apr-2009.) |
β’ βπ¦(π₯ β π¦ β§ βπ§(π§ β π¦ β βπ€(π§ β π€ β§ π€ β π¦))) | ||
Theorem | inf5 9514 | The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see Theorem infeq5 9506). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.) |
β’ βπ₯ π₯ β βͺ π₯ | ||
Theorem | omelon 9515 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
β’ Ο β On | ||
Theorem | dfom3 9516* | The class of natural numbers Ο can be defined as the intersection of all inductive sets (which is the smallest inductive set, since inductive sets are closed under intersection), which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.) |
β’ Ο = β© {π₯ β£ (β β π₯ β§ βπ¦ β π₯ suc π¦ β π₯)} | ||
Theorem | elom3 9517* | A simplification of elom 7795 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) |
β’ (π΄ β Ο β βπ₯(Lim π₯ β π΄ β π₯)) | ||
Theorem | dfom4 9518* | A simplification of df-om 7793 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) |
β’ Ο = {π₯ β£ βπ¦(Lim π¦ β π₯ β π¦)} | ||
Theorem | dfom5 9519 | Ο is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.) |
β’ Ο = β© {π₯ β£ Lim π₯} | ||
Theorem | oancom 9520 | Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.) |
β’ (1o +o Ο) β (Ο +o 1o) | ||
Theorem | isfinite 9521 | A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.) |
β’ (π΄ β Fin β π΄ βΊ Ο) | ||
Theorem | fict 9522 | A finite set is countable (weaker version of isfinite 9521). (Contributed by Thierry Arnoux, 27-Mar-2018.) |
β’ (π΄ β Fin β π΄ βΌ Ο) | ||
Theorem | nnsdom 9523 | A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.) |
β’ (π΄ β Ο β π΄ βΊ Ο) | ||
Theorem | omenps 9524 | Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.) |
β’ Ο β (Ο β {β }) | ||
Theorem | omensuc 9525 | The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
β’ Ο β suc Ο | ||
Theorem | infdifsn 9526 | Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.) |
β’ (Ο βΌ π΄ β (π΄ β {π΅}) β π΄) | ||
Theorem | infdiffi 9527 | Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) |
β’ ((Ο βΌ π΄ β§ π΅ β Fin) β (π΄ β π΅) β π΄) | ||
Theorem | unbnn3 9528* | Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 9176 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.) |
β’ ((π΄ β Ο β§ βπ₯ β Ο βπ¦ β π΄ π₯ β π¦) β π΄ β Ο) | ||
Theorem | noinfep 9529* | Using the Axiom of Regularity in the form zfregfr 9474, show that there are no infinite descending β-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.) |
β’ βπ₯ β Ο (πΉβsuc π₯) β (πΉβπ₯) | ||
Syntax | ccnf 9530 | Extend class notation with the Cantor normal form function. |
class CNF | ||
Definition | df-cnf 9531* | Define the Cantor normal form function, which takes as input a finitely supported function from π¦ to π₯ and outputs the corresponding member of the ordinal exponential π₯ βo π¦. The content of the original Cantor Normal Form theorem is that for π₯ = Ο this function is a bijection onto Ο βo π¦ for any ordinal π¦ (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 9564 of this function in terms of df-oi 9379. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ CNF = (π₯ β On, π¦ β On β¦ (π β {π β (π₯ βm π¦) β£ π finSupp β } β¦ β¦OrdIso( E , (π supp β )) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π₯ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β )βdom β))) | ||
Theorem | cantnffval 9532* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = {π β (π΄ βm π΅) β£ π finSupp β } & β’ (π β π΄ β On) & β’ (π β π΅ β On) β β’ (π β (π΄ CNF π΅) = (π β π β¦ β¦OrdIso( E , (π supp β )) / ββ¦(seqΟ((π β V, π§ β V β¦ (((π΄ βo (ββπ)) Β·o (πβ(ββπ))) +o π§)), β )βdom β))) | ||
Theorem | cantnfdm 9533* | The domain of the Cantor normal form function (in later lemmas we will use dom (π΄ CNF π΅) to abbreviate "the set of finitely supported functions from π΅ to π΄"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = {π β (π΄ βm π΅) β£ π finSupp β } & β’ (π β π΄ β On) & β’ (π β π΅ β On) β β’ (π β dom (π΄ CNF π΅) = π) | ||
Theorem | cantnfvalf 9534* | Lemma for cantnf 9562. The function appearing in cantnfval 9537 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.) |
β’ πΉ = seqΟ((π β π΄, π§ β π΅ β¦ (πΆ +o π·)), β ) β β’ πΉ:ΟβΆOn | ||
Theorem | cantnfs 9535 | Elementhood in the set of finitely supported functions from π΅ to π΄. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) β β’ (π β (πΉ β π β (πΉ:π΅βΆπ΄ β§ πΉ finSupp β ))) | ||
Theorem | cantnfcl 9536 | Basic properties of the order isomorphism πΊ used later. The support of an πΉ β π is a finite subset of π΄, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ (π β πΉ β π) β β’ (π β ( E We (πΉ supp β ) β§ dom πΊ β Ο)) | ||
Theorem | cantnfval 9537* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ (π β πΉ β π) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) +o π§)), β ) β β’ (π β ((π΄ CNF π΅)βπΉ) = (π»βdom πΊ)) | ||
Theorem | cantnfval2 9538* | Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ (π β πΉ β π) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) +o π§)), β ) β β’ (π β ((π΄ CNF π΅)βπΉ) = (seqΟ((π β dom πΊ, π§ β On β¦ (((π΄ βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) +o π§)), β )βdom πΊ)) | ||
Theorem | cantnfsuc 9539* | The value of the recursive function π» at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ (π β πΉ β π) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) +o π§)), β ) β β’ ((π β§ πΎ β Ο) β (π»βsuc πΎ) = (((π΄ βo (πΊβπΎ)) Β·o (πΉβ(πΊβπΎ))) +o (π»βπΎ))) | ||
Theorem | cantnfle 9540* | A lower bound on the CNF function. Since ((π΄ CNF π΅)βπΉ) is defined as the sum of (π΄ βo π₯) Β·o (πΉβπ₯) over all π₯ in the support of πΉ, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all πΆ β π΅ instead of just those πΆ in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ (π β πΉ β π) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) +o π§)), β ) & β’ (π β πΆ β π΅) β β’ (π β ((π΄ βo πΆ) Β·o (πΉβπΆ)) β ((π΄ CNF π΅)βπΉ)) | ||
Theorem | cantnflt 9541* | An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent π΄ βo πΆ where πΆ is larger than any exponent (πΊβπ₯), π₯ β πΎ which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ (π β πΉ β π) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) +o π§)), β ) & β’ (π β β β π΄) & β’ (π β πΎ β suc dom πΊ) & β’ (π β πΆ β On) & β’ (π β (πΊ β πΎ) β πΆ) β β’ (π β (π»βπΎ) β (π΄ βo πΆ)) | ||
Theorem | cantnflt2 9542 | An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β πΉ β π) & β’ (π β β β π΄) & β’ (π β πΆ β On) & β’ (π β (πΉ supp β ) β πΆ) β β’ (π β ((π΄ CNF π΅)βπΉ) β (π΄ βo πΆ)) | ||
Theorem | cantnff 9543 | The CNF function is a function from finitely supported functions from π΅ to π΄, to the ordinal exponential π΄ βo π΅. (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) β β’ (π β (π΄ CNF π΅):πβΆ(π΄ βo π΅)) | ||
Theorem | cantnf0 9544 | The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β β β π΄) β β’ (π β ((π΄ CNF π΅)β(π΅ Γ {β })) = β ) | ||
Theorem | cantnfrescl 9545* | A function is finitely supported from π΅ to π΄ iff the extended function is finitely supported from π· to π΄. (Contributed by Mario Carneiro, 25-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β π· β On) & β’ (π β π΅ β π·) & β’ ((π β§ π β (π· β π΅)) β π = β ) & β’ (π β β β π΄) & β’ π = dom (π΄ CNF π·) β β’ (π β ((π β π΅ β¦ π) β π β (π β π· β¦ π) β π)) | ||
Theorem | cantnfres 9546* | The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β π· β On) & β’ (π β π΅ β π·) & β’ ((π β§ π β (π· β π΅)) β π = β ) & β’ (π β β β π΄) & β’ π = dom (π΄ CNF π·) & β’ (π β (π β π΅ β¦ π) β π) β β’ (π β ((π΄ CNF π΅)β(π β π΅ β¦ π)) = ((π΄ CNF π·)β(π β π· β¦ π))) | ||
Theorem | cantnfp1lem1 9547* | Lemma for cantnfp1 9550. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β πΊ β π) & β’ (π β π β π΅) & β’ (π β π β π΄) & β’ (π β (πΊ supp β ) β π) & β’ πΉ = (π‘ β π΅ β¦ if(π‘ = π, π, (πΊβπ‘))) β β’ (π β πΉ β π) | ||
Theorem | cantnfp1lem2 9548* | Lemma for cantnfp1 9550. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β πΊ β π) & β’ (π β π β π΅) & β’ (π β π β π΄) & β’ (π β (πΊ supp β ) β π) & β’ πΉ = (π‘ β π΅ β¦ if(π‘ = π, π, (πΊβπ‘))) & β’ (π β β β π) & β’ π = OrdIso( E , (πΉ supp β )) β β’ (π β dom π = suc βͺ dom π) | ||
Theorem | cantnfp1lem3 9549* | Lemma for cantnfp1 9550. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β πΊ β π) & β’ (π β π β π΅) & β’ (π β π β π΄) & β’ (π β (πΊ supp β ) β π) & β’ πΉ = (π‘ β π΅ β¦ if(π‘ = π, π, (πΊβπ‘))) & β’ (π β β β π) & β’ π = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πβπ)) Β·o (πΉβ(πβπ))) +o π§)), β ) & β’ πΎ = OrdIso( E , (πΊ supp β )) & β’ π = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πΎβπ)) Β·o (πΊβ(πΎβπ))) +o π§)), β ) β β’ (π β ((π΄ CNF π΅)βπΉ) = (((π΄ βo π) Β·o π) +o ((π΄ CNF π΅)βπΊ))) | ||
Theorem | cantnfp1 9550* | If πΉ is created by adding a single term (πΉβπ) = π to πΊ, where π is larger than any element of the support of πΊ, then πΉ is also a finitely supported function and it is assigned the value ((π΄ βo π) Β·o π) +o π§ where π§ is the value of πΊ. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ (π β πΊ β π) & β’ (π β π β π΅) & β’ (π β π β π΄) & β’ (π β (πΊ supp β ) β π) & β’ πΉ = (π‘ β π΅ β¦ if(π‘ = π, π, (πΊβπ‘))) β β’ (π β (πΉ β π β§ ((π΄ CNF π΅)βπΉ) = (((π΄ βo π) Β·o π) +o ((π΄ CNF π΅)βπΊ)))) | ||
Theorem | oemapso 9551* | The relation π is a strict order on π (a corollary of wemapso2 9422). (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} β β’ (π β π Or π) | ||
Theorem | oemapval 9552* | Value of the relation π. (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) β β’ (π β (πΉππΊ β βπ§ β π΅ ((πΉβπ§) β (πΊβπ§) β§ βπ€ β π΅ (π§ β π€ β (πΉβπ€) = (πΊβπ€))))) | ||
Theorem | oemapvali 9553* | If πΉ < πΊ, then there is some π§ witnessing this, but we can say more and in fact there is a definable expression π that also witnesses πΉ < πΊ. (Contributed by Mario Carneiro, 25-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β πΉππΊ) & β’ π = βͺ {π β π΅ β£ (πΉβπ) β (πΊβπ)} β β’ (π β (π β π΅ β§ (πΉβπ) β (πΊβπ) β§ βπ€ β π΅ (π β π€ β (πΉβπ€) = (πΊβπ€)))) | ||
Theorem | cantnflem1a 9554* | Lemma for cantnf 9562. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β πΉππΊ) & β’ π = βͺ {π β π΅ β£ (πΉβπ) β (πΊβπ)} β β’ (π β π β (πΊ supp β )) | ||
Theorem | cantnflem1b 9555* | Lemma for cantnf 9562. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β πΉππΊ) & β’ π = βͺ {π β π΅ β£ (πΉβπ) β (πΊβπ)} & β’ π = OrdIso( E , (πΊ supp β )) β β’ ((π β§ (suc π’ β dom π β§ (β‘πβπ) β π’)) β π β (πβπ’)) | ||
Theorem | cantnflem1c 9556* | Lemma for cantnf 9562. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β πΉππΊ) & β’ π = βͺ {π β π΅ β£ (πΉβπ) β (πΊβπ)} & β’ π = OrdIso( E , (πΊ supp β )) β β’ ((((π β§ (suc π’ β dom π β§ (β‘πβπ) β π’)) β§ π₯ β π΅) β§ ((πΉβπ₯) β β β§ (πβπ’) β π₯)) β π₯ β (πΊ supp β )) | ||
Theorem | cantnflem1d 9557* | Lemma for cantnf 9562. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β πΉππΊ) & β’ π = βͺ {π β π΅ β£ (πΉβπ) β (πΊβπ)} & β’ π = OrdIso( E , (πΊ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πβπ)) Β·o (πΊβ(πβπ))) +o π§)), β ) β β’ (π β ((π΄ CNF π΅)β(π₯ β π΅ β¦ if(π₯ β π, (πΉβπ₯), β ))) β (π»βsuc (β‘πβπ))) | ||
Theorem | cantnflem1 9558* | Lemma for cantnf 9562. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation π is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct πΉ, πΊ are π -related as πΉ < πΊ or πΊ < πΉ, and WLOG assuming that πΉ < πΊ, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β πΉππΊ) & β’ π = βͺ {π β π΅ β£ (πΉβπ) β (πΊβπ)} & β’ π = OrdIso( E , (πΊ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (((π΄ βo (πβπ)) Β·o (πΊβ(πβπ))) +o π§)), β ) β β’ (π β ((π΄ CNF π΅)βπΉ) β ((π΄ CNF π΅)βπΊ)) | ||
Theorem | cantnflem2 9559* | Lemma for cantnf 9562. (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΆ β (π΄ βo π΅)) & β’ (π β πΆ β ran (π΄ CNF π΅)) & β’ (π β β β πΆ) β β’ (π β (π΄ β (On β 2o) β§ πΆ β (On β 1o))) | ||
Theorem | cantnflem3 9560* | Lemma for cantnf 9562. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than πΆ has a normal form, we can use oeeu 8517 to factor πΆ into the form ((π΄ βo π) Β·o π) +o π where 0 < π < π΄ and π < (π΄ βo π) (and a fortiori π < π΅). Then since π < (π΄ βo π) β€ (π΄ βo π) Β·o π β€ πΆ, π has a normal form, and by appending the term (π΄ βo π) Β·o π using cantnfp1 9550 we get a normal form for πΆ. (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΆ β (π΄ βo π΅)) & β’ (π β πΆ β ran (π΄ CNF π΅)) & β’ (π β β β πΆ) & β’ π = βͺ β© {π β On β£ πΆ β (π΄ βo π)} & β’ π = (β©πβπ β On βπ β (π΄ βo π)(π = β¨π, πβ© β§ (((π΄ βo π) Β·o π) +o π) = πΆ)) & β’ π = (1st βπ) & β’ π = (2nd βπ) & β’ (π β πΊ β π) & β’ (π β ((π΄ CNF π΅)βπΊ) = π) & β’ πΉ = (π‘ β π΅ β¦ if(π‘ = π, π, (πΊβπ‘))) β β’ (π β πΆ β ran (π΄ CNF π΅)) | ||
Theorem | cantnflem4 9561* | Lemma for cantnf 9562. Complete the induction step of cantnflem3 9560. (Contributed by Mario Carneiro, 25-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} & β’ (π β πΆ β (π΄ βo π΅)) & β’ (π β πΆ β ran (π΄ CNF π΅)) & β’ (π β β β πΆ) & β’ π = βͺ β© {π β On β£ πΆ β (π΄ βo π)} & β’ π = (β©πβπ β On βπ β (π΄ βo π)(π = β¨π, πβ© β§ (((π΄ βo π) Β·o π) +o π) = πΆ)) & β’ π = (1st βπ) & β’ π = (2nd βπ) β β’ (π β πΆ β ran (π΄ CNF π΅)) | ||
Theorem | cantnf 9562* | The Cantor Normal Form theorem. The function (π΄ CNF π΅), which maps a finitely supported function from π΅ to π΄ to the sum ((π΄ βo π(π1)) β π1) +o ((π΄ βo π(π2)) β π2) +o ... over all indices π < π΅ such that π(π) is nonzero, is an order isomorphism from the ordering π of finitely supported functions to the set (π΄ βo π΅) under the natural order. Setting π΄ = Ο and letting π΅ be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 9546, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} β β’ (π β (π΄ CNF π΅) Isom π, E (π, (π΄ βo π΅))) | ||
Theorem | oemapwe 9563* | The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} β β’ (π β (π We π β§ dom OrdIso(π, π) = (π΄ βo π΅))) | ||
Theorem | cantnffval2 9564* | An alternate definition of df-cnf 9531 which relies on cantnf 9562. (Note that although the use of π seems self-referential, one can use cantnfdm 9533 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) & β’ π = {β¨π₯, π¦β© β£ βπ§ β π΅ ((π₯βπ§) β (π¦βπ§) β§ βπ€ β π΅ (π§ β π€ β (π₯βπ€) = (π¦βπ€)))} β β’ (π β (π΄ CNF π΅) = β‘OrdIso(π, π)) | ||
Theorem | cantnff1o 9565 | Simplify the isomorphism of cantnf 9562 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.) |
β’ π = dom (π΄ CNF π΅) & β’ (π β π΄ β On) & β’ (π β π΅ β On) β β’ (π β (π΄ CNF π΅):πβ1-1-ontoβ(π΄ βo π΅)) | ||
Theorem | wemapwe 9566* | Construct lexicographic order on a function space based on a reverse well-ordering of the indices and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.) |
β’ π = {β¨π₯, π¦β© β£ βπ§ β π΄ ((π₯βπ§)π(π¦βπ§) β§ βπ€ β π΄ (π§π π€ β (π₯βπ€) = (π¦βπ€)))} & β’ π = {π₯ β (π΅ βm π΄) β£ π₯ finSupp π} & β’ (π β π We π΄) & β’ (π β π We π΅) & β’ (π β π΅ β β ) & β’ πΉ = OrdIso(π , π΄) & β’ πΊ = OrdIso(π, π΅) & β’ π = (πΊββ ) β β’ (π β π We π) | ||
Theorem | oef1o 9567* | A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (πΉββ ) = β can be discharged using fveqf1o 7243.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
β’ (π β πΉ:π΄β1-1-ontoβπΆ) & β’ (π β πΊ:π΅β1-1-ontoβπ·) & β’ (π β π΄ β (On β 1o)) & β’ (π β π΅ β On) & β’ (π β πΆ β On) & β’ (π β π· β On) & β’ (π β (πΉββ ) = β ) & β’ πΎ = (π¦ β {π₯ β (π΄ βm π΅) β£ π₯ finSupp β } β¦ (πΉ β (π¦ β β‘πΊ))) & β’ π» = (((πΆ CNF π·) β πΎ) β β‘(π΄ CNF π΅)) β β’ (π β π»:(π΄ βo π΅)β1-1-ontoβ(πΆ βo π·)) | ||
Theorem | cnfcomlem 9568* | Lemma for cnfcom 9569. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ (π β π΄ β On) & β’ (π β π΅ β (Ο βo π΄)) & β’ πΉ = (β‘(Ο CNF π΄)βπ΅) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ (π β πΌ β dom πΊ) & β’ (π β π β (Ο βo (πΊβπΌ))) & β’ (π β (πβπΌ):(π»βπΌ)β1-1-ontoβπ) β β’ (π β (πβsuc πΌ):(π»βsuc πΌ)β1-1-ontoβ((Ο βo (πΊβπΌ)) Β·o (πΉβ(πΊβπΌ)))) | ||
Theorem | cnfcom 9569* | Any ordinal π΅ is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ (π β π΄ β On) & β’ (π β π΅ β (Ο βo π΄)) & β’ πΉ = (β‘(Ο CNF π΄)βπ΅) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ (π β πΌ β dom πΊ) β β’ (π β (πβsuc πΌ):(π»βsuc πΌ)β1-1-ontoβ((Ο βo (πΊβπΌ)) Β·o (πΉβ(πΊβπΌ)))) | ||
Theorem | cnfcom2lem 9570* | Lemma for cnfcom2 9571. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ (π β π΄ β On) & β’ (π β π΅ β (Ο βo π΄)) & β’ πΉ = (β‘(Ο CNF π΄)βπ΅) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ π = (πΊββͺ dom πΊ) & β’ (π β β β π΅) β β’ (π β dom πΊ = suc βͺ dom πΊ) | ||
Theorem | cnfcom2 9571* | Any nonzero ordinal π΅ is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ (π β π΄ β On) & β’ (π β π΅ β (Ο βo π΄)) & β’ πΉ = (β‘(Ο CNF π΄)βπ΅) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ π = (πΊββͺ dom πΊ) & β’ (π β β β π΅) β β’ (π β (πβdom πΊ):π΅β1-1-ontoβ((Ο βo π) Β·o (πΉβπ))) | ||
Theorem | cnfcom3lem 9572* | Lemma for cnfcom3 9573. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ (π β π΄ β On) & β’ (π β π΅ β (Ο βo π΄)) & β’ πΉ = (β‘(Ο CNF π΄)βπ΅) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ π = (πΊββͺ dom πΊ) & β’ (π β Ο β π΅) β β’ (π β π β (On β 1o)) | ||
Theorem | cnfcom3 9573* | Any infinite ordinal π΅ is equinumerous to a power of Ο. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 9575.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ (π β π΄ β On) & β’ (π β π΅ β (Ο βo π΄)) & β’ πΉ = (β‘(Ο CNF π΄)βπ΅) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ π = (πΊββͺ dom πΊ) & β’ (π β Ο β π΅) & β’ π = (π’ β (πΉβπ), π£ β (Ο βo π) β¦ (((πΉβπ) Β·o π£) +o π’)) & β’ π = (π’ β (πΉβπ), π£ β (Ο βo π) β¦ (((Ο βo π) Β·o π’) +o π£)) & β’ π = ((π β β‘π) β (πβdom πΊ)) β β’ (π β π:π΅β1-1-ontoβ(Ο βo π)) | ||
Theorem | cnfcom3clem 9574* | Lemma for cnfcom3c 9575. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.) |
β’ π = dom (Ο CNF π΄) & β’ πΉ = (β‘(Ο CNF π΄)βπ) & β’ πΊ = OrdIso( E , (πΉ supp β )) & β’ π» = seqΟ((π β V, π§ β V β¦ (π +o π§)), β ) & β’ π = seqΟ((π β V, π β V β¦ πΎ), β ) & β’ π = ((Ο βo (πΊβπ)) Β·o (πΉβ(πΊβπ))) & β’ πΎ = ((π₯ β π β¦ (dom π +o π₯)) βͺ β‘(π₯ β dom π β¦ (π +o π₯))) & β’ π = (πΊββͺ dom πΊ) & β’ π = (π’ β (πΉβπ), π£ β (Ο βo π) β¦ (((πΉβπ) Β·o π£) +o π’)) & β’ π = (π’ β (πΉβπ), π£ β (Ο βo π) β¦ (((Ο βo π) Β·o π’) +o π£)) & β’ π = ((π β β‘π) β (πβdom πΊ)) & β’ πΏ = (π β (Ο βo π΄) β¦ π) β β’ (π΄ β On β βπβπ β π΄ (Ο β π β βπ€ β (On β 1o)(πβπ):πβ1-1-ontoβ(Ο βo π€))) | ||
Theorem | cnfcom3c 9575* | Wrap the construction of cnfcom3 9573 into an existential quantifier. For any Ο β π, there is a bijection from π to some power of Ο. Furthermore, this bijection is canonical , which means that we can find a single function π which will give such bijections for every π less than some arbitrarily large bound π΄. (Contributed by Mario Carneiro, 30-May-2015.) |
β’ (π΄ β On β βπβπ β π΄ (Ο β π β βπ€ β (On β 1o)(πβπ):πβ1-1-ontoβ(Ο βo π€))) | ||
Syntax | cttrcl 9576 | Declare the syntax for the transitive closure of a class. |
class t++π | ||
Definition | df-ttrcl 9577* | Define the transitive closure of a class. This is the smallest relationship containing π (or more precisely, the relation (π βΎ V) induced by π ) and having the transitive property. Definition from [Levy] p. 59, who denotes it as π β and calls it the "ancestral" of π . (Contributed by Scott Fenton, 17-Oct-2024.) |
β’ t++π = {β¨π₯, π¦β© β£ βπ β (Ο β 1o)βπ(π Fn suc π β§ ((πββ ) = π₯ β§ (πβπ) = π¦) β§ βπ β π (πβπ)π (πβsuc π))} | ||
Theorem | ttrcleq 9578 | Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) |
β’ (π = π β t++π = t++π) | ||
Theorem | nfttrcld 9579 | Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.) |
β’ (π β β²π₯π ) β β’ (π β β²π₯t++π ) | ||
Theorem | nfttrcl 9580 | Bound variable hypothesis builder for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) |
β’ β²π₯π β β’ β²π₯t++π | ||
Theorem | relttrcl 9581 | The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.) |
β’ Rel t++π | ||
Theorem | brttrcl 9582* | Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.) |
β’ (π΄t++π π΅ β βπ β (Ο β 1o)βπ(π Fn suc π β§ ((πββ ) = π΄ β§ (πβπ) = π΅) β§ βπ β π (πβπ)π (πβsuc π))) | ||
Theorem | brttrcl2 9583* | Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.) |
β’ (π΄t++π π΅ β βπ β Ο βπ(π Fn suc suc π β§ ((πββ ) = π΄ β§ (πβsuc π) = π΅) β§ βπ β suc π(πβπ)π (πβsuc π))) | ||
Theorem | ssttrcl 9584 | If π is a relation, then it is a subclass of its transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.) |
β’ (Rel π β π β t++π ) | ||
Theorem | ttrcltr 9585 | The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.) |
β’ (t++π β t++π ) β t++π | ||
Theorem | ttrclresv 9586 | The transitive closure of π restricted to V is the same as the transitive closure of π itself. (Contributed by Scott Fenton, 20-Oct-2024.) |
β’ t++(π βΎ V) = t++π | ||
Theorem | ttrclco 9587 | Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.) |
β’ (t++π β π ) β t++π | ||
Theorem | cottrcl 9588 | Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.) |
β’ (π β t++π ) β t++π | ||
Theorem | ttrclss 9589 | If π is a subclass of π and π is transitive, then the transitive closure of π is a subclass of π. (Contributed by Scott Fenton, 20-Oct-2024.) |
β’ ((π β π β§ (π β π) β π) β t++π β π) | ||
Theorem | dmttrcl 9590 | The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.) |
β’ dom t++π = dom π | ||
Theorem | rnttrcl 9591 | The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.) |
β’ ran t++π = ran π | ||
Theorem | ttrclexg 9592 | If π is a set, then so is t++π . (Contributed by Scott Fenton, 26-Oct-2024.) |
β’ (π β π β t++π β V) | ||
Theorem | dfttrcl2 9593* | When π is a set and a relationship, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.) |
β’ ((π β π β§ Rel π ) β t++π = β© {π§ β£ (π β π§ β§ (π§ β π§) β π§)}) | ||
Theorem | ttrclselem1 9594* | Lemma for ttrclse 9596. Show that all finite ordinal function values of πΉ are subsets of π΄. (Contributed by Scott Fenton, 31-Oct-2024.) |
β’ πΉ = rec((π β V β¦ βͺ π€ β π Pred(π , π΄, π€)), Pred(π , π΄, π)) β β’ (π β Ο β (πΉβπ) β π΄) | ||
Theorem | ttrclselem2 9595* | Lemma for ttrclse 9596. Show that a suc π element long chain gives membership in the π-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.) |
β’ πΉ = rec((π β V β¦ βͺ π€ β π Pred(π , π΄, π€)), Pred(π , π΄, π)) β β’ ((π β Ο β§ π Se π΄ β§ π β π΄) β (βπ(π Fn suc suc π β§ ((πββ ) = π¦ β§ (πβsuc π) = π) β§ βπ β suc π(πβπ)(π βΎ π΄)(πβsuc π)) β π¦ β (πΉβπ))) | ||
Theorem | ttrclse 9596 |
If π
is set-like over π΄, then
the transitive closure of the
restriction of π
to π΄ is set-like over π΄.
This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.) |
β’ (π Se π΄ β t++(π βΎ π΄) Se π΄) | ||
Theorem | trcl 9597* | For any set π΄, show the properties of its transitive closure πΆ. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9598 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
β’ π΄ β V & β’ πΉ = (rec((π§ β V β¦ (π§ βͺ βͺ π§)), π΄) βΎ Ο) & β’ πΆ = βͺ π¦ β Ο (πΉβπ¦) β β’ (π΄ β πΆ β§ Tr πΆ β§ βπ₯((π΄ β π₯ β§ Tr π₯) β πΆ β π₯)) | ||
Theorem | tz9.1 9598* |
Every set has a transitive closure (the smallest transitive extension).
Theorem 9.1 of [TakeutiZaring] p.
73. See trcl 9597 for an explicit
expression for the transitive closure. Apparently open problems are
whether this theorem can be proved without the Axiom of Infinity; if
not, then whether it implies Infinity; and if not, what is the
"property" that Infinity has that the other axioms don't have
that is
weaker than Infinity itself?
(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.) |
β’ π΄ β V β β’ βπ₯(π΄ β π₯ β§ Tr π₯ β§ βπ¦((π΄ β π¦ β§ Tr π¦) β π₯ β π¦)) | ||
Theorem | tz9.1c 9599* | Alternate expression for the existence of transitive closures tz9.1 9598: the intersection of all transitive sets containing π΄ is a set. (Contributed by Mario Carneiro, 22-Mar-2013.) |
β’ π΄ β V β β’ β© {π₯ β£ (π΄ β π₯ β§ Tr π₯)} β V | ||
Theorem | epfrs 9600* | The strong form of the Axiom of Regularity (no sethood requirement on π΄), with the axiom itself present as an antecedent. See also zfregs 9601. (Contributed by Mario Carneiro, 22-Mar-2013.) |
β’ (( E Fr π΄ β§ π΄ β β ) β βπ₯ β π΄ (π₯ β© π΄) = β ) |
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