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

Theoremoiiso2 7501 The order isomorphism of the well-order on is an isomorphism onto (which is a subset of by oif 7500). (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremordtype 7502 For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoiiniseg 7503 is an initial segment of under the well-order . (Contributed by Mario Carneiro, 26-Jun-2015.)
OrdIso        Se

Theoremordtype2 7504 For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto isomorphically. Otherwise, is a proper class, which implies that either is a proper class or . This weak version of ordtype 7502 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoiexg 7505 The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso

Theoremoion 7506 The order type of the well-order on is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoiiso 7507 The order isomorphism of the well-order on is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoien 7508 The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoieu 7509 Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoismo 7510 When is a subclass of , is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of ). The proof avoids ax-rep 4321 (the second statement is trivial under ax-rep 4321). (Contributed by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoremoiid 7511 The order type of an ordinal under the order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoremhartogslem1 7512* Lemma for hartogs 7514. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
OrdIso

Theoremhartogslem2 7513* Lemma for hartogs 7514. (Contributed by Mario Carneiro, 14-Jan-2013.)
OrdIso

Theoremhartogs 7514* Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 8439- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremwofib 7515 The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)

Theoremwemaplem1 7516* Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwemaplem2 7517* Lemma for wemapso 7521. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremwemaplem3 7518* Lemma for wemapso 7521. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremwemappo 7519* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwemapso2lem 7520* Lemma for wemapso 7521. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremwemapso 7521* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremwemapso2 7522* An alternative to having a well-order on in wemapso 7521 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremcard2on 7523* Proof that the alternate definition cardval2 7879 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)

Theoremcard2inf 7524* The definition cardval2 7879 has the curious property that for non-numerable sets (for which ndmfv 5756 yields ), it still evaluates to a non-empty set, and indeed it contains . (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

2.4.40  Hartogs function, order types, weak dominance

Syntaxchar 7525 Class symbol for the Hartogs/cardinal successor function.
har

Syntaxcwdom 7526 Class symbol for the weak dominance relation.
*

Definitiondf-har 7527* Define the Hartogs function , which maps all sets to the smallest ordinal that cannot be injected into the given set. In the important special case where is an ordinal, this is the cardinal successor operation.

Traditionally, the Hartogs number of a set is written and the cardinal successor ; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 7828.

Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.)

har

Definitiondf-wdom 7528* A set is weakly dominated by a "larger" set iff the "larger" set can be mapped onto the "smaller" set or the smaller set is empty; equivalently if the smaller set can be placed into bijection with some partition of the larger set. When choice is assumed (as fodom 8403), this concides with the 1-1 defition df-dom 7112; however, it is not known whether this is a choice-equivalent or a strictly weaker form. Some discussion of this question can be found at http://boolesrings.org/asafk/2014/on-the-partition-principle/. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremharf 7529 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har

Theoremharcl 7530 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har

Theoremharval 7531* Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har

Theoremelharval 7532 The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
har

Theoremharndom 7533 The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
har

Theoremharword 7534 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
har har

Theoremrelwdom 7535 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdom 7536* Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdomi 7537* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
*

Theorembrwdomn0 7538* Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theorem0wdom 7539 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremfowdom 7540 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremwdomref 7541 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdom2 7542* 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 7543 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremwdomtr 7544 Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
* * *

Theoremwdomen1 7545 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
* *

Theoremwdomen2 7546 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
* *

Theoremwdompwdom 7547 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 7548 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7261, equivalent to canth 6540). (Contributed by Mario Carneiro, 15-May-2015.)
*

Theoremwdom2d 7549* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4321). (Contributed by Stefan O'Rear, 13-Feb-2015.)
*

Theoremwdomd 7550* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
*

Theorembrwdom3 7551* Condition for weak dominance with a condition reminiscent of wdomd 7550. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
*

Theorembrwdom3i 7552* Weak dominance implies existance of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
*

Theoremunwdomg 7553 Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
* * *

Theoremxpwdomg 7554 Weak dominance of a cross product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
* * *

Theoremwdomima2g 7555 A set is weakly dominant over its image under any function. This version of wdomimag 7556 is stated so as to avoid ax-rep 4321. (Contributed by Mario Carneiro, 25-Jun-2015.)
*

Theoremwdomimag 7556 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 7557 Lemma for unxpwdom 7558. (Contributed by Mario Carneiro, 15-May-2015.)
*

Theoremunxpwdom 7558 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 7559 The Hartogs function is weakly dominated by . This follows from a more precise analysis of the bound used in hartogs 7514 to prove that har is a set. (Contributed by Mario Carneiro, 15-May-2015.)
har *

Theoremixpiunwdom 7560* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7093 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 7561* 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 7564) 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 7566). A stronger version that works for proper classes is proved as zfregs 7669. (Contributed by NM, 14-Aug-1993.)

Theoremaxreg2 7562* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)

Theoremzfregcl 7563* The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.)

Theoremzfreg 7564* 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 7669). (Contributed by NM, 26-Nov-1995.)

Theoremzfreg2 7565* The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7564) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)

Theoremelirrv 7566 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 7571 and efrirr 4564, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)

Theoremelirr 7567 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 7568 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 7569 The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)

TheoremruALT 7570 Alternate proof of Russell's Paradox ru 3161, simplified using (indirectly) the Axiom of Regularity ax-reg 7561. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremzfregfr 7571 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)

Theoremen2lp 7572 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 7573 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)

Theoremopthreg 7574 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7561 (via the preleq 7573 step). See df-op 3824 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)

Theoremsuc11reg 7575 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 7576* Assuming ax-reg 7561, 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 7577* 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 7594. (Contributed by NM, 15-Oct-1996.)

Theoreminf1 7578 Variation of Axiom of Infinity (using zfinf 7595 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)

Theoreminf2 7579* Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7595 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lema 7580* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lemb 7581* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lemc 7582* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lemd 7583* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lem1 7584* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lem2 7585* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 28-Oct-1996.)

Theoreminf3lem3 7586* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7564. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem4 7587* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem5 7588* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem6 7589* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. (Contributed by NM, 29-Oct-1996.)

Theoreminf3lem7 7590* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7591 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 5972. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)

Theoreminf3 7591 Our Axiom of Infinity ax-inf 7594 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7579, 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 7596 and zfinf2 7598.) The main proof is provided by inf3lema 7580 through inf3lem7 7590, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7590. 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 7581.)
F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 7582.)
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 7584.)
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 7585.)
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 7586.)
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 7587.)
Proof:  Lemmas 1 and 3.

Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 7588.)
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 7589.)
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 7590.)
Q.E.D.

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

Theoreminfeq5i 7592 Half of infeq5 7593. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoreminfeq5 7593 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 7599.) 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 7594* 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 7578 and inf2 7579). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7598 and omex 7599 and are based on the (nontrivial) proof of inf3 7591. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7597. Theorem inf0 7577 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7601 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7597 requires this axiom along with Regularity ax-reg 7561 for its derivation (as theorem axinf2 7596 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7597 instead of this one. The derivation of this axiom from ax-inf2 7597 is shown by theorem axinf 7600.

Proofs should normally use the standard version ax-inf2 7597 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

Theoremzfinf 7595* Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)

Theoremaxinf2 7596* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 7594 and Regularity ax-reg 7561.

This theorem should not be referenced in any proof. Instead, use ax-inf2 7597 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 7597* 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 7598 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7596 above, using our version of Infinity ax-inf 7594 and the Axiom of Regularity ax-reg 7561. We will reference ax-inf2 7597 instead of axinf2 7596 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7594 from ax-inf2 7597 is shown by theorem axinf 7600. (Contributed by NM, 30-Aug-1993.)

Theoremzfinf2 7598* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7597 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)

2.6.2  Existence of omega (the set of natural numbers)

Theoremomex 7599 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 7577.

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 4857 and (the universe of all sets) by fineqv 7325. 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 4865 through peano5 4869 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

Theoremaxinf 7600* The first version of the Axiom of Infinity ax-inf 7594 proved from the second version ax-inf2 7597. Note that we didn't use ax-reg 7561, unlike the other direction axinf2 7596. (Contributed by NM, 24-Apr-2009.)

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