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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembrwdomn0 7501* Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theorem0wdom 7502 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremfowdom 7503 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremwdomref 7504 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdom2 7505* Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremdomwdom 7506 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremwdomtr 7507 Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
* * *

Theoremwdomen1 7508 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
* *

Theoremwdomen2 7509 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
* *

Theoremwdompwdom 7510 Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theoremcanthwdom 7511 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7227, equivalent to canth 6506). (Contributed by Mario Carneiro, 15-May-2015.)
*

Theoremwdom2d 7512* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4288). (Contributed by Stefan O'Rear, 13-Feb-2015.)
*

Theoremwdomd 7513* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
*

Theorembrwdom3 7514* Condition for weak dominance with a condition reminiscent of wdomd 7513. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
*

Theorembrwdom3i 7515* Weak dominance implies existance of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
*

Theoremunwdomg 7516 Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
* * *

Theoremxpwdomg 7517 Weak dominance of a cross product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
* * *

Theoremwdomima2g 7518 A set is weakly dominant over its image under any function. This version of wdomimag 7519 is stated so as to avoid ax-rep 4288. (Contributed by Mario Carneiro, 25-Jun-2015.)
*

Theoremwdomimag 7519 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.)
*

Theoremunxpwdom2 7520 Lemma for unxpwdom 7521. (Contributed by Mario Carneiro, 15-May-2015.)
*

Theoremunxpwdom 7521 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.)
*

Theoremharwdom 7522 The Hartogs function is weakly dominated by . This follows from a more precise analysis of the bound used in hartogs 7477 to prove that har is a set. (Contributed by Mario Carneiro, 15-May-2015.)
har *

Theoremixpiunwdom 7523* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7059 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

Axiomax-reg 7524* 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 7527) 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 7529). A stronger version that works for proper classes is proved as zfregs 7632. (Contributed by NM, 14-Aug-1993.)

Theoremaxreg2 7525* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)

Theoremzfregcl 7526* The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.)

Theoremzfreg 7527* 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 7632). (Contributed by NM, 26-Nov-1995.)

Theoremzfreg2 7528* The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7527) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)

Theoremelirrv 7529 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 7534 and efrirr 4531, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)

Theoremelirr 7530 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.)

Theoremsucprcreg 7531 A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.)

Theoremruv 7532 The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)

TheoremruALT 7533 Alternate proof of Russell's Paradox ru 3128, simplified using (indirectly) the Axiom of Regularity ax-reg 7524. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremzfregfr 7534 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)

Theoremen2lp 7535 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.)

Theorempreleq 7536 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)

Theoremopthreg 7537 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7524 (via the preleq 7536 step). See df-op 3791 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)

Theoremsuc11reg 7538 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.)

Theoremdford2 7539* Assuming ax-reg 7524, 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

Theoreminf0 7540* 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 7557. (Contributed by NM, 15-Oct-1996.)

Theoreminf1 7541 Variation of Axiom of Infinity (using zfinf 7558 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)

Theoreminf2 7542* Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7558 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lema 7543* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lemb 7544* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lemc 7545* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lemd 7546* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lem1 7547* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lem2 7548* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lem3 7549* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7527. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem4 7550* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem5 7551* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem6 7552* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem7 7553* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7554 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 5938. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)

Theoreminf3 7554 Our Axiom of Infinity ax-inf 7557 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7542, 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 7559 and zfinf2 7561.) The main proof is provided by inf3lema 7543 through inf3lem7 7553, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7553. 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 7544.)
F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 7545.)
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 7547.)
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 7548.)
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 7549.)
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 7550.)
Proof:  Lemmas 1 and 3.

Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 7551.)
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 7552.)
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 7553.)
Q.E.D.

(Contributed by NM, 29-Oct-1996.)

Theoreminfeq5i 7555 Half of infeq5 7556. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoreminfeq5 7556 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 7562.) 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.)

2.6  ZF Set Theory - add the Axiom of Infinity

2.6.1  Introduce the Axiom of Infinity

Axiomax-inf 7557* 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 7541 and inf2 7542). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7561 and omex 7562 and are based on the (nontrivial) proof of inf3 7554. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7560. Theorem inf0 7540 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7564 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7560 requires this axiom along with Regularity ax-reg 7524 for its derivation (as theorem axinf2 7559 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7560 instead of this one. The derivation of this axiom from ax-inf2 7560 is shown by theorem axinf 7563.

Proofs should normally use the standard version ax-inf2 7560 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

Theoremzfinf 7558* Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)

Theoremaxinf2 7559* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 7557 and Regularity ax-reg 7524.

This theorem should not be referenced in any proof. Instead, use ax-inf2 7560 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)

Axiomax-inf2 7560* 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 7561 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7559 above, using our version of Infinity ax-inf 7557 and the Axiom of Regularity ax-reg 7524. We will reference ax-inf2 7560 instead of axinf2 7559 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7557 from ax-inf2 7560 is shown by theorem axinf 7563. (Contributed by NM, 30-Aug-1993.)

Theoremzfinf2 7561* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7560 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)

2.6.2  Existence of omega (the set of natural numbers)

Theoremomex 7562 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 7540.

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 4823 and (the universe of all sets) by fineqv 7291. 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 4831 through peano5 4835 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

Theoremaxinf 7563* The first version of the Axiom of Infinity ax-inf 7557 proved from the second version ax-inf2 7560. Note that we didn't use ax-reg 7524, unlike the other direction axinf2 7559. (Contributed by NM, 24-Apr-2009.)

Theoreminf5 7564 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 7556). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)

Theoremomelon 7565 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)

Theoremdfom3 7566* 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.)

Theoremelom3 7567* A simplification of elom 4815 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)

Theoremdfom4 7568* A simplification of df-om 4813 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)

Theoremdfom5 7569 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.)

Theoremoancom 7570 Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)

Theoremisfinite 7571 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.)

Theoremnnsdom 7572 A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)

Theoremomenps 7573 Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)

Theoremomensuc 7574 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)

Theoreminfdifsn 7575 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.)

Theoreminfdiffi 7576 Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremunbnn3 7577* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 7330 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)

Theoremnoinfep 7578* Using the Axiom of Regularity in the form zfregfr 7534, 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.)

TheoremnoinfepOLD 7579* Using the Axiom of Regularity in the form zfregfr 7534, 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

Syntaxccnf 7580 Extend class notation with the Cantor normal form function.
CNF

Definitiondf-cnf 7581* 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 7615 of this function in terms of df-oi 7443. (Contributed by Mario Carneiro, 25-May-2015.)
CNF OrdIso seq𝜔

Theoremcantnffval 7582* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
CNF OrdIso seq𝜔

Theoremcantnfdm 7583* 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

Theoremcantnfvalf 7584* Lemma for cantnf 7613. The function appearing in cantnfval 7587 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
seq𝜔

Theoremcantnfs 7585 Elementhood in the set of finitely supported functions from to . (Contributed by Mario Carneiro, 25-May-2015.)
CNF

Theoremcantnfcl 7586 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

Theoremcantnfval 7587* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso               seq𝜔        CNF

Theoremcantnfval2 7588* Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso               seq𝜔        CNF seq𝜔

Theoremcantnfsuc 7589* The value of the recursive function at a successor. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                      OrdIso               seq𝜔

Theoremcantnfle 7590* 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               seq𝜔               CNF

Theoremcantnflt 7591* 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               seq𝜔

Theoremcantnflt2 7592 An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                  CNF

Theoremcantnff 7593 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

Theoremcantnf0 7594 The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
CNF                             CNF

Theoremcantnfreslem 7595* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.)
CNF

Theoremcantnfrescl 7596* 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

Theoremcantnfres 7597* The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
CNF                                                  CNF               CNF CNF

Theoremcantnfp1lem1 7598* Lemma for cantnfp1 7601. (Contributed by Mario Carneiro, 20-Jun-2015.)
CNF

Theoremcantnfp1lem2 7599* Lemma for cantnfp1 7601. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                                OrdIso

Theoremcantnfp1lem3 7600* Lemma for cantnfp1 7601. (Contributed by Mario Carneiro, 28-May-2015.)
CNF                                                                OrdIso        seq𝜔        OrdIso        seq𝜔        CNF CNF

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