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Type | Label | Description |
---|---|---|

Statement | ||

Theorem | wdomimag 7301 | A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |

^{*} | ||

Theorem | unxpwdom2 7302 | Lemma for unxpwdom 7303. (Contributed by Mario Carneiro, 15-May-2015.) |

^{*} | ||

Theorem | unxpwdom 7303 | If a cross product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.) |

^{*} | ||

Theorem | harwdom 7304 | The Hartogs function is weakly dominated by . This follows from a more precise analysis of the bound used in hartogs 7259 to prove that har is a set. (Contributed by Mario Carneiro, 15-May-2015.) |

har ^{*}
| ||

Theorem | ixpiunwdom 7305* | Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 6846 this shows that and have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.) |

^{*} | ||

2.5 ZF Set Theory - add the Axiom of
Regularity | ||

2.5.1 Introduce the Axiom of
Regularity | ||

Axiom | ax-reg 7306* | Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 7309) that every non-empty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 7311). A stronger version that works for proper classes is proved as zfregs 7414. (Contributed by NM, 14-Aug-1993.) |

Theorem | axreg2 7307* | Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.) |

Theorem | zfregcl 7308* | The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) |

Theorem | zfreg 7309* | The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." There is also a "strong form," not requiring that be a set, that can be proved with more difficulty (see zfregs 7414). (Contributed by NM, 26-Nov-1995.) |

Theorem | zfreg2 7310* | The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7309) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.) |

Theorem | elirrv 7311 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 7316 and efrirr 4374, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.) |

Theorem | elirr 7312 | No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | sucprcreg 7313 | A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) |

Theorem | ruv 7314 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |

Theorem | ruALT 7315 | Alternate proof of Russell's Paradox ru 2990, simplified using (indirectly) the Axiom of Regularity ax-reg 7306. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |

Theorem | zfregfr 7316 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |

Theorem | en2lp 7317 | No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.) |

Theorem | preleq 7318 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |

Theorem | opthreg 7319 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7306 (via the preleq 7318 step). See df-op 3649 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |

Theorem | suc11reg 7320 | The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |

Theorem | dford2 7321* | Assuming ax-reg 7306, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.) |

2.5.2 Axiom of Infinity equivalents | ||

Theorem | inf0 7322* | Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class " " exists, is a subset of its union, and contains a given set (and thus is non-empty). Thus it provides an example demonstrating that a set exists with the necessary properties demanded by ax-inf 7339. (Contributed by NM, 15-Oct-1996.) |

Theorem | inf1 7323 | Variation of Axiom of Infinity (using zfinf 7340 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.) |

Theorem | inf2 7324* | Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7340 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.) |

Theorem | inf3lema 7325* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 28-Oct-1996.) |

Theorem | inf3lemb 7326* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 28-Oct-1996.) |

Theorem | inf3lemc 7327* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 28-Oct-1996.) |

Theorem | inf3lemd 7328* | |

Theorem | inf3lem1 7329* | |

Theorem | inf3lem2 7330* | |

Theorem | inf3lem3 7331* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7309. (Contributed by NM, 29-Oct-1996.) |

Theorem | inf3lem4 7332* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 29-Oct-1996.) |

Theorem | inf3lem5 7333* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 29-Oct-1996.) |

Theorem | inf3lem6 7334* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. (Contributed by NM, 29-Oct-1996.) |

Theorem | inf3lem7 7335* | Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 5751. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.) |

Theorem | inf3 7336 |
Our Axiom of Infinity ax-inf 7339 implies the standard Axiom of Infinity.
The hypothesis is a variant of our Axiom of Infinity provided by
inf2 7324, 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 7341 and zfinf2 7343.) The main proof is provided by
inf3lema 7325 through inf3lem7 7335, and this final piece eliminates the
auxiliary hypothesis of inf3lem7 7335. 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 non-empty 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 7326.) F_n+1 = {y<X | y^X subset F_n} (See inf3lemc 7327.) 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 7329.) 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 7330.) 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 7331.) 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 7332.) Proof: Lemmas 1 and 3. Lemma 5. F_m proper_subset F_n, m < n. (See inf3lem5 7333.) 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 7334.) 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 7335.) Q.E.D.(Contributed by NM, 29-Oct-1996.) |

Theorem | infeq5i 7337 | Half of infeq5 7338. (Contributed by Mario Carneiro, 16-Nov-2014.) |

Theorem | infeq5 7338 | 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 7344.) The left-hand side provides us with a very short way to express of 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.) |

2.6 ZF Set Theory - add the Axiom of
Infinity | ||

2.6.1 Introduce the Axiom of
Infinity | ||

Axiom | ax-inf 7339* |
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 , an infinite set
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 7323
and
inf2 7324). More standard versions, which essentially
state that there
exists a set containing all the natural numbers, are shown as zfinf2 7343
and omex 7344 and are based on the (nontrivial) proof of inf3 7336.
This
version has the advantage that when expanded to primitives, it has fewer
symbols than the standard version ax-inf2 7342. Theorem inf0 7322
shows the
reverse derivation of our axiom from a standard one. Theorem inf5 7346
shows a very short way to state this axiom.
The standard version of Infinity ax-inf2 7342 requires this axiom along with Regularity ax-reg 7306 for its derivation (as theorem axinf2 7341 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7342 instead of this one. The derivation of this axiom from ax-inf2 7342 is shown by theorem axinf 7345. Proofs should normally use the standard version ax-inf2 7342 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.) |

Theorem | zfinf 7340* | Axiom of Infinity expressed with fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |

Theorem | axinf2 7341* |
A standard version of Axiom of Infinity, expanded to primitives, derived
from our version of Infinity ax-inf 7339 and Regularity ax-reg 7306.
This theorem should not be referenced in any proof. Instead, use ax-inf2 7342 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.) |

Axiom | ax-inf2 7342* | 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 7343 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7341 above, using our version of Infinity ax-inf 7339 and the Axiom of Regularity ax-reg 7306. We will reference ax-inf2 7342 instead of axinf2 7341 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7339 from ax-inf2 7342 is shown by theorem axinf 7345. (Contributed by NM, 30-Aug-1993.) |

Theorem | zfinf2 7343* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7342 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.) |

2.6.2 Existence of omega (the set of natural
numbers) | ||

Theorem | omex 7344 |
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 7322.
A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ; this would lead to by omon 4667 and (the universe of all sets) by fineqv 7078. 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 4675 through peano5 4679 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.) |

Theorem | axinf 7345* | The first version of the Axiom of Infinity ax-inf 7339 proved from the second version ax-inf2 7342. Note that we didn't use ax-reg 7306, unlike the other direction axinf2 7341. (Contributed by NM, 24-Apr-2009.) |

Theorem | inf5 7346 | The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 7338). This provides us with a very compact way to express of the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.) |

Theorem | omelon 7347 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |

Theorem | dfom3 7348* | The class of natural numbers omega can be defined as the smallest "inductive set," which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. (Contributed by NM, 6-Aug-1994.) |

Theorem | elom3 7349* | A simplification of elom 4659 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) |

Theorem | dfom4 7350* | A simplification of df-om 4657 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.) |

Theorem | dfom5 7351 | 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.) |

Theorem | oancom 7352 | Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.) |

Theorem | isfinite 7353 | 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.) |

Theorem | nnsdom 7354 | A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.) |

Theorem | omenps 7355 | Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.) |

Theorem | omensuc 7356 | The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |

Theorem | infdifsn 7357 | 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.) |

Theorem | infdiffi 7358 | Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) |

Theorem | unbnn3 7359* | Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 7113 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.) |

Theorem | noinfep 7360* | Using the Axiom of Regularity in the form zfregfr 7316, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.) |

Theorem | noinfepOLD 7361* | Using the Axiom of Regularity in the form zfregfr 7316, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |

2.6.3 Cantor normal form | ||

Syntax | ccnf 7362 | Extend class notation with the Cantor normal form function. |

CNF | ||

Definition | df-cnf 7363* | Define the Cantor normal form function, which takes as input a finitely supported function from to and outputs the corresponding member of the ordinal exponential . The content of the original Cantor Normal Form theorem is that for this function is a bijection onto for any ordinal (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to ). 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 7397 of this function in terms of df-oi 7225. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF
OrdIso seq_{𝜔}
| ||

Theorem | cantnffval 7364* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF OrdIso seq_{𝜔}
| ||

Theorem | cantnfdm 7365* | The domain of the Cantor normal form function (in later lemmas we will use CNF to abbreviate "the set of finitely supported functions from to "). (Contributed by Mario Carneiro, 25-May-2015.) |

CNF | ||

Theorem | cantnfvalf 7366* | Lemma for cantnf 7395. The function appearing in cantnfval 7369 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.) |

seq_{𝜔} | ||

Theorem | cantnfs 7367 | Elementhood in the set of finitely supported functions from to . (Contributed by Mario Carneiro, 25-May-2015.) |

CNF | ||

Theorem | cantnfcl 7368 | Basic properties of the order isomorphism used later. The support of an is a finite subset of , so it is well-ordered by and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF OrdIso | ||

Theorem | cantnfval 7369* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF OrdIso
seq_{𝜔} CNF | ||

Theorem | cantnfval2 7370* | Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF OrdIso
seq_{𝜔} CNF seq_{𝜔} | ||

Theorem | cantnfsuc 7371* | The value of the recursive function at a successor. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF OrdIso
seq_{𝜔} | ||

Theorem | cantnfle 7372* | A lower bound on the CNF function. Since CNF is defined as the sum of over all in the support of , it is larger than any of these terms (and all other terms are zero so we can extend the statement to all instead of just those in the support). (Contributed by Mario Carneiro, 28-May-2015.) |

CNF OrdIso
seq_{𝜔} CNF | ||

Theorem | cantnflt 7373* | 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 where is larger than any exponent which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF OrdIso
seq_{𝜔}
| ||

Theorem | cantnflt2 7374 | An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF | ||

Theorem | cantnff 7375 | The CNF function is a function from finitely supported functions from to , to the ordinal exponential . (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF | ||

Theorem | cantnf0 7376 | The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.) |

CNF CNF | ||

Theorem | cantnfreslem 7377* | The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF | ||

Theorem | cantnfrescl 7378* | A function is finitely supported from to iff the extended function is finitely supported from to . (Contributed by Mario Carneiro, 25-May-2015.) |

CNF CNF | ||

Theorem | cantnfres 7379* | The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF CNF CNF CNF | ||

Theorem | cantnfp1lem1 7380* | Lemma for cantnfp1 7383. (Contributed by Mario Carneiro, 20-Jun-2015.) |

CNF | ||

Theorem | cantnfp1lem2 7381* | Lemma for cantnfp1 7383. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF OrdIso | ||

Theorem | cantnfp1lem3 7382* | Lemma for cantnfp1 7383. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF OrdIso seq_{𝜔} OrdIso
seq_{𝜔} CNF
CNF
| ||

Theorem | cantnfp1 7383* | If is created by adding a single term to , where is larger than any element of the support of , then is also a finitely supported function and it is assigned the value where is the value of . (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF CNF | ||

Theorem | oemapso 7384* | The relation is a strict order on (a corollary of wemapso2 7267). (Contributed by Mario Carneiro, 28-May-2015.) |

CNF | ||

Theorem | oemapval 7385* | Value of the relation . (Contributed by Mario Carneiro, 28-May-2015.) |

CNF | ||

Theorem | oemapvali 7386* | If , then there is some witnessing this, but we can say more and in fact there is a definable expression that also witnesses . (Contributed by Mario Carneiro, 25-May-2015.) |

CNF | ||

Theorem | cantnflem1a 7387* | Lemma for cantnf 7395. (Contributed by Mario Carneiro, 4-Jun-2015.) |

CNF | ||

Theorem | cantnflem1b 7388* | Lemma for cantnf 7395. (Contributed by Mario Carneiro, 4-Jun-2015.) |

CNF OrdIso | ||

Theorem | cantnflem1c 7389* | Lemma for cantnf 7395. (Contributed by Mario Carneiro, 4-Jun-2015.) |

CNF OrdIso | ||

Theorem | cantnflem1d 7390* | Lemma for cantnf 7395. (Contributed by Mario Carneiro, 4-Jun-2015.) |

CNF
OrdIso
seq_{𝜔} CNF
| ||

Theorem | cantnflem1 7391* | Lemma for cantnf 7395. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 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 are -related as or , and WLOG assuming that , we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF
OrdIso
seq_{𝜔} CNF CNF | ||

Theorem | cantnflem2 7392* | Lemma for cantnf 7395. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF | ||

Theorem | cantnflem3 7393* | Lemma for cantnf 7395. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than has a normal form, we can use oeeu 6601 to factor into the form where and (and a fortiori ). Then since , has a normal form, and by appending the term using cantnfp1 7383 we get a normal form for . (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF CNF CNF | ||

Theorem | cantnflem4 7394* | Lemma for cantnf 7395. Complete the induction step of cantnflem3 7393. (Contributed by Mario Carneiro, 25-May-2015.) |

CNF CNF CNF | ||

Theorem | cantnf 7395* | The Cantor Normal Form theorem. The function CNF , which maps a finitely supported function from to to the sum over all indexes such that is nonzero, is an order isomorphism from the ordering of finitely supported functions to the set under the natural order. Setting and letting be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 7379, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF | ||

Theorem | oemapwe 7396* | The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternative definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) |

CNF OrdIso | ||

Theorem | cantnffval2 7397* | An alternative definition of df-cnf 7363 which relies on cantnf 7395. (Note that although the use of seems self-referential, one can use cantnfdm 7365 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) |

CNF CNF OrdIso | ||

Theorem | cantnff1o 7398 | Simplify the isomorphism of cantnf 7395 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.) |

CNF CNF | ||

Theorem | mapfien 7399* | A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) |

Theorem | wemapwe 7400* | Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) |

OrdIso OrdIso |

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