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Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremen2lp 7201 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.)
 |- 
 -.  ( A  e.  B  /\  B  e.  A )
 
Theorempreleq 7202 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( ( A  e.  B  /\  C  e.  D )  /\  { A ,  B }  =  { C ,  D } )  ->  ( A  =  C  /\  B  =  D ) )
 
Theoremopthreg 7203 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7190 (via the preleq 7202 step). See df-op 3553 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremsuc11reg 7204 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.)
 |-  ( suc  A  =  suc  B  <->  A  =  B )
 
Theoremdford2 7205* Assuming ax-reg 7190, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
 |-  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) )
 
2.5.2  Axiom of Infinity equivalents
 
Theoreminf0 7206* Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class " ran  ( rec (
( v  e.  _V  |->  suc  v ) ,  x
)  |`  om ) " exists, is a subset of its union, and contains a given set  x (and thus is non-empty). Thus it provides an example demonstrating that a set  y exists with the necessary properties demanded by ax-inf 7223. (Contributed by NM, 15-Oct-1996.)
 |- 
 om  e.  _V   =>    |- 
 E. y ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y
 ) ) )
 
Theoreminf1 7207 Variation of Axiom of Infinity (using zfinf 7224 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
 |- 
 E. x ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )   =>    |-  E. x ( x  =/=  (/)  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )
 
Theoreminf2 7208* Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7224 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
 |- 
 E. x ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )   =>    |-  E. x ( x  =/=  (/)  /\  x  C_ 
 U. x )
 
Theoreminf3lema 7209* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  ( G `  B )  <->  ( A  e.  x  /\  ( A  i^i  x )  C_  B )
 )
 
Theoreminf3lemb 7210* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F `  (/) )  =  (/)
 
Theoreminf3lemc 7211* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  om  ->  ( F `  suc  A )  =  ( G `  ( F `  A ) ) )
 
Theoreminf3lemd 7212* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  om  ->  ( F `  A )  C_  x )
 
Theoreminf3lem1 7213* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
 
Theoreminf3lem2 7214* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 28-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( A  e.  om  ->  ( F `  A )  =/=  x ) )
 
Theoreminf3lem3 7215* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7193. (Contributed by NM, 29-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( A  e.  om  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
 
Theoreminf3lem4 7216* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 29-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( A  e.  om  ->  ( F `  A ) 
 C.  ( F `  suc  A ) ) )
 
Theoreminf3lem5 7217* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 29-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  ( ( A  e.  om 
 /\  B  e.  A )  ->  ( F `  B )  C.  ( F `
  A ) ) )
 
Theoreminf3lem6 7218* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. (Contributed by NM, 29-Oct-1996.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  F : om -1-1-> ~P x )
 
Theoreminf3lem7 7219* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7220 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 5603. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
 |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )   &    |-  F  =  ( rec ( G ,  (/) )  |`  om )   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( x  =/=  (/)  /\  x  C_ 
 U. x )  ->  om  e.  _V )
 
Theoreminf3 7220 Our Axiom of Infinity ax-inf 7223 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7208, 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 7225 and zfinf2 7227.) The main proof is provided by inf3lema 7209 through inf3lem7 7219, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7219. 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 7210.)
       F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 7211.)
       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 7213.)
       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 7214.)
       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 7215.)
       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 7216.)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 7217.)
       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 7218.)
       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 7219.)
       Q.E.D.
       
(Contributed by NM, 29-Oct-1996.)
 |- 
 E. x ( x  =/=  (/)  /\  x  C_  U. x )   =>    |- 
 om  e.  _V
 
Theoreminfeq5i 7221 Half of infeq5 7222. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( om  e.  _V  ->  E. x  x  C.  U. x )
 
Theoreminfeq5 7222 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 7228.) 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.)
 |-  ( E. x  x 
 C.  U. x  <->  om  e.  _V )
 
2.6  ZF Set Theory - add the Axiom of Infinity
 
2.6.1  Introduce the Axiom of Infinity
 
Axiomax-inf 7223* 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  x, an infinite set  y 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 7207 and inf2 7208). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7227 and omex 7228 and are based on the (nontrivial) proof of inf3 7220. Our version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7226. Theorem inf0 7206 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7230 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7226 requires this axiom along with Regularity ax-reg 7190 for its derivation (as theorem axinf2 7225 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7226 instead of this one. The derivation of this axiom from ax-inf2 7226 is shown by theorem axinf 7229. (Contributed by NM, 16-Aug-1993.)

 |- 
 E. y ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y
 ) ) )
 
Theoremzfinf 7224* Axiom of Infinity expressed with fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
 |- 
 E. x ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) )
 
Theoremaxinf2 7225* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 7223 and Regularity ax-reg 7190.

This theorem should not be referenced in any proof. Instead, use ax-inf2 7226 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)

 |- 
 E. x ( E. y ( y  e.  x  /\  A. z  -.  z  e.  y
 )  /\  A. y ( y  e.  x  ->  E. z ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) ) )
 
Axiomax-inf2 7226* 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 7227 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7225 above, using our version of Infinity ax-inf 7223 and the Axiom of Regularity ax-reg 7190. We will reference ax-inf2 7226 instead of axinf2 7225 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7223 from ax-inf2 7226 is shown by theorem axinf 7229. (Contributed by NM, 30-Aug-1993.)
 |- 
 E. x ( E. y ( y  e.  x  /\  A. z  -.  z  e.  y
 )  /\  A. y ( y  e.  x  ->  E. z ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) ) )
 
Theoremzfinf2 7227* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7226 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
 |- 
 E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
 
2.6.2  Existence of omega (the set of natural numbers)
 
Theoremomex 7228 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 7206.

A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial  -.  om  e.  _V; this would lead to  om  =  On by omon 4558 and  Fin  =  _V (the universe of all sets) by fineqv 6963. 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 4566 through peano5 4570 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

 |- 
 om  e.  _V
 
Theoremaxinf 7229* The first version of the Axiom of Infinity ax-inf 7223 proved from the second version ax-inf2 7226. Note that we didn't use ax-reg 7190, unlike the other direction axinf2 7225. (Contributed by NM, 24-Apr-2009.)
 |- 
 E. y ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y
 ) ) )
 
Theoreminf5 7230 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 7222). 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.)
 |- 
 E. x  x  C.  U. x
 
Theoremomelon 7231 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
 |- 
 om  e.  On
 
Theoremdfom3 7232* 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.)
 |- 
 om  =  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
 
Theoremelom3 7233* A simplification of elom 4550 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
 |-  ( A  e.  om  <->  A. x ( Lim  x  ->  A  e.  x ) )
 
Theoremdfom4 7234* A simplification of df-om 4548 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
 |- 
 om  =  { x  |  A. y ( Lim  y  ->  x  e.  y ) }
 
Theoremdfom5 7235  om 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.)
 |- 
 om  =  |^| { x  |  Lim  x }
 
Theoremoancom 7236 Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)
 |-  ( 1o  +o  om )  =/=  ( om  +o  1o )
 
Theoremisfinite 7237 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.)
 |-  ( A  e.  Fin  <->  A  ~<  om )
 
Theoremnnsdom 7238 A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)
 |-  ( A  e.  om  ->  A  ~<  om )
 
Theoremomenps 7239 Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)
 |- 
 om  ~~  ( om  \  { (/) } )
 
Theoremomensuc 7240 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |- 
 om  ~~  suc  om
 
Theoreminfdifsn 7241 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.)
 |-  ( om  ~<_  A  ->  ( A  \  { B } )  ~~  A )
 
Theoreminfdiffi 7242 Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B  e.  Fin )  ->  ( A  \  B ) 
 ~~  A )
 
Theoremunbnn3 7243* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 6998 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
 |-  ( ( A  C_  om 
 /\  A. x  e.  om  E. y  e.  A  x  e.  y )  ->  A  ~~ 
 om )
 
Theoremnoinfep 7244* Using the Axiom of Regularity in the form zfregfr 7200, show that there are no infinite descending 
e.-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
 |- 
 E. x  e.  om  ( F `  suc  x )  e/  ( F `  x )
 
TheoremnoinfepOLD 7245* Using the Axiom of Regularity in the form zfregfr 7200, show that there are no infinite descending 
e.-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( F  Fn  om  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `
  x ) )
 
2.6.3  Cantor normal form
 
Syntaxccnf 7246 Extend class notation with the Cantor normal form function.
 class CNF
 
Definitiondf-cnf 7247* Define the Cantor normal form function, which takes as input a finitely supported function from  y to  x and outputs the corresponding member of the ordinal exponential  x  ^o  y. The content of the original Cantor Normal Form theorem is that for  x  =  om this function is a bijection onto  om  ^o  y for any ordinal  y (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 7281 of this function in terms of df-oi 7109. (Contributed by Mario Carneiro, 25-May-2015.)
 |- CNF 
 =  ( x  e. 
 On ,  y  e. 
 On  |->  ( f  e. 
 { g  e.  ( x  ^m  y )  |  ( `' g "
 ( _V  \  1o ) )  e.  Fin } 
 |->  [_OrdIso (  _E  ,  ( `' f " ( _V  \  1o ) ) ) 
 /  h ]_ (seq𝜔 (
 ( k  e.  _V ,  z  e.  _V  |->  ( ( ( x 
 ^o  ( h `  k ) )  .o  ( f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h ) ) )
 
Theoremcantnffval 7248* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_OrdIso
 (  _E  ,  ( `' f " ( _V  \  1o ) ) ) 
 /  h ]_ (seq𝜔 (
 ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A 
 ^o  ( h `  k ) )  .o  ( f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h ) ) )
 
Theoremcantnfdm 7249* The domain of the Cantor normal form function (in later lemmas we will use  dom  ( A CNF 
B ) to abbreviate "the set of finitely supported functions from  B to  A"). (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  dom  (  A CNF  B )  =  S )
 
Theoremcantnfvalf 7250* Lemma for cantnf 7279. The function appearing in cantnfval 7253 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  F  = seq𝜔 ( ( k  e.  A ,  z  e.  B  |->  ( C  +o  D ) ) ,  (/) )   =>    |-  F : om --> On
 
Theoremcantnfs 7251 Elementhood in the set of finitely supported functions from  B to  A. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  ( F  e.  S  <->  ( F : B
 --> A  /\  ( `' F " ( _V  \  1o ) )  e. 
 Fin ) ) )
 
Theoremcantnfcl 7252 Basic properties of the order isomorphism  G used later. The support of an  F  e.  S is a finite subset of  A, so it is well-ordered by  _E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  G  = OrdIso (  _E 
 ,  ( `' F " ( _V  \  1o ) ) )   &    |-  ( ph  ->  F  e.  S )   =>    |-  ( ph  ->  (  _E  We  ( `' F " ( _V  \  1o ) )  /\  dom  G  e.  om ) )
 
Theoremcantnfval 7253* The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  G  = OrdIso (  _E 
 ,  ( `' F " ( _V  \  1o ) ) )   &    |-  ( ph  ->  F  e.  S )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( G `
  k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  F )  =  ( H `  dom  G ) )
 
Theoremcantnfval2 7254* Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  G  = OrdIso (  _E 
 ,  ( `' F " ( _V  \  1o ) ) )   &    |-  ( ph  ->  F  e.  S )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( G `
  k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  F )  =  (seq𝜔 ( ( k  e. 
 dom  G ,  z  e. 
 On  |->  ( ( ( A  ^o  ( G `
  k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
 ) ) ,  (/) ) `  dom  G ) )
 
Theoremcantnfsuc 7255* The value of the recursive function 
H at a successor. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  G  = OrdIso (  _E 
 ,  ( `' F " ( _V  \  1o ) ) )   &    |-  ( ph  ->  F  e.  S )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( G `
  k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ( ph  /\  K  e.  om )  ->  ( H `  suc  K )  =  ( ( ( A  ^o  ( G `  K ) )  .o  ( F `  ( G `  K ) ) )  +o  ( H `  K ) ) )
 
Theoremcantnfle 7256* A lower bound on the CNF function. Since  ( ( A CNF 
B ) `  F
) is defined as the sum of  ( A  ^o  x )  .o  ( F `  x ) over all  x in the support of  F, it is larger than any of these terms (and all other terms are zero so we can extend the statement to all  C  e.  B instead of just those  C in the support). (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  G  = OrdIso (  _E 
 ,  ( `' F " ( _V  \  1o ) ) )   &    |-  ( ph  ->  F  e.  S )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( G `
  k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
 ) ) ,  (/) )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  (
 ( A  ^o  C )  .o  ( F `  C ) )  C_  ( ( A CNF  B ) `  F ) )
 
Theoremcantnflt 7257* 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  A  ^o  C where  C is larger than any exponent  ( G `  x ) ,  x  e.  K which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  G  = OrdIso (  _E 
 ,  ( `' F " ( _V  \  1o ) ) )   &    |-  ( ph  ->  F  e.  S )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( G `
  k ) )  .o  ( F `  ( G `  k ) ) )  +o  z
 ) ) ,  (/) )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ph  ->  K  e.  suc 
 dom  G )   &    |-  ( ph  ->  C  e.  On )   &    |-  ( ph  ->  ( G " K )  C_  C )   =>    |-  ( ph  ->  ( H `  K )  e.  ( A  ^o  C ) )
 
Theoremcantnflt2 7258 An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ph  ->  C  e.  On )   &    |-  ( ph  ->  ( `' F " ( _V  \  1o ) )  C_  C )   =>    |-  ( ph  ->  ( ( A CNF  B ) `  F )  e.  ( A  ^o  C ) )
 
Theoremcantnff 7259 The CNF function is a function from finitely supported functions from  B to  A, to the ordinal exponential  A  ^o  B. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
 
Theoremcantnf0 7260 The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  (/)  e.  A )   =>    |-  ( ph  ->  (
 ( A CNF  B ) `  ( B  X.  { (/)
 } ) )  =  (/) )
 
Theoremcantnfreslem 7261* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  D  e.  On )   &    |-  ( ph  ->  B 
 C_  D )   &    |-  (
 ( ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )   =>    |-  ( ph  ->  ( `' ( n  e.  B  |->  X ) " ( _V  \  1o ) )  =  ( `' ( n  e.  D  |->  X )
 " ( _V  \  1o ) ) )
 
Theoremcantnfrescl 7262* A function is finitely supported from  B to  A iff the extended function is finitely supported from  D to  A. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  D  e.  On )   &    |-  ( ph  ->  B 
 C_  D )   &    |-  (
 ( ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  T  =  dom  (  A CNF  D )   =>    |-  ( ph  ->  ( ( n  e.  B  |->  X )  e.  S  <->  ( n  e.  D  |->  X )  e.  T ) )
 
Theoremcantnfres 7263* The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  D  e.  On )   &    |-  ( ph  ->  B 
 C_  D )   &    |-  (
 ( ph  /\  n  e.  ( D  \  B ) )  ->  X  =  (/) )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  T  =  dom  (  A CNF  D )   &    |-  ( ph  ->  ( n  e.  B  |->  X )  e.  S )   =>    |-  ( ph  ->  (
 ( A CNF  B ) `  ( n  e.  B  |->  X ) )  =  ( ( A CNF  D ) `  ( n  e.  D  |->  X ) ) )
 
Theoremcantnfp1lem1 7264* Lemma for cantnfp1 7267. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  ( `' G " ( _V  \  1o ) )  C_  X )   &    |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `
  t ) ) )   =>    |-  ( ph  ->  F  e.  S )
 
Theoremcantnfp1lem2 7265* Lemma for cantnfp1 7267. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  ( `' G " ( _V  \  1o ) )  C_  X )   &    |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `
  t ) ) )   &    |-  ( ph  ->  (/)  e.  Y )   &    |-  O  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   =>    |-  ( ph  ->  dom 
 O  =  suc  U. dom  O )
 
Theoremcantnfp1lem3 7266* Lemma for cantnfp1 7267. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  ( `' G " ( _V  \  1o ) )  C_  X )   &    |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `
  t ) ) )   &    |-  ( ph  ->  (/)  e.  Y )   &    |-  O  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( O `
  k ) )  .o  ( F `  ( O `  k ) ) )  +o  z
 ) ) ,  (/) )   &    |-  K  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   &    |-  M  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( K `
  k ) )  .o  ( G `  ( K `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  F )  =  ( ( ( A 
 ^o  X )  .o  Y )  +o  (
 ( A CNF  B ) `  G ) ) )
 
Theoremcantnfp1 7267* If  F is created by adding a single term  ( F `  X
)  =  Y to  G, where  X is larger than any element of the support of  G, then  F is also a finitely supported function and it is assigned the value  ( ( A  ^o  X )  .o  Y
)  +o  z where  z is the value of  G. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  ( `' G " ( _V  \  1o ) )  C_  X )   &    |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `
  t ) ) )   =>    |-  ( ph  ->  ( F  e.  S  /\  ( ( A CNF  B ) `  F )  =  ( ( ( A 
 ^o  X )  .o  Y )  +o  (
 ( A CNF  B ) `  G ) ) ) )
 
Theoremoemapso 7268* The relation  T is a strict order on  S (a corollary of wemapso2 7151). (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  T  Or  S )
 
Theoremoemapval 7269* Value of the relation  T. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   =>    |-  ( ph  ->  ( F T G  <->  E. z  e.  B  ( ( F `  z )  e.  ( G `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) ) )
 
Theoremoemapvali 7270* If  F  <  G, then there is some  z witnessing this, but we can say more and in fact there is a definable expression  X that also witnesses  F  <  G. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   =>    |-  ( ph  ->  ( X  e.  B  /\  ( F `  X )  e.  ( G `  X )  /\  A. w  e.  B  ( X  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) )
 
Theoremcantnflem1a 7271* Lemma for cantnf 7279. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   =>    |-  ( ph  ->  X  e.  ( `' G " ( _V  \  1o ) ) )
 
Theoremcantnflem1b 7272* Lemma for cantnf 7279. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   =>    |-  ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u ) )  ->  X  C_  ( O `  u ) )
 
Theoremcantnflem1c 7273* Lemma for cantnf 7279. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   =>    |-  ( ( ( (
 ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
 C_  u ) ) 
 /\  x  e.  B )  /\  ( ( F `
  x )  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  ( `' G "
 ( _V  \  1o ) ) )
 
Theoremcantnflem1d 7274* Lemma for cantnf 7279. (Contributed by Mario Carneiro, 4-Jun-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( O `
  k ) )  .o  ( G `  ( O `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  ( x  e.  B  |->  if ( x  C_  X ,  ( F `  x ) ,  (/) ) ) )  e.  ( H `
  suc  ( `' O `  X ) ) )
 
Theoremcantnflem1 7275* Lemma for cantnf 7279. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation  T 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  F ,  G are  T -related as  F  <  G or  G  <  F, and WLOG assuming that  F  <  G, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  F  e.  S )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  F T G )   &    |-  X  =  U. { c  e.  B  |  ( F `  c )  e.  ( G `  c ) }   &    |-  O  = OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( ( ( A  ^o  ( O `
  k ) )  .o  ( G `  ( O `  k ) ) )  +o  z
 ) ) ,  (/) )   =>    |-  ( ph  ->  ( ( A CNF  B ) `
  F )  e.  ( ( A CNF  B ) `  G ) )
 
Theoremcantnflem2 7276* Lemma for cantnf 7279. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  C  e.  ( A  ^o  B ) )   &    |-  ( ph  ->  C  C_  ran  (  A CNF  B ) )   &    |-  ( ph  ->  (/)  e.  C )   =>    |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
 
Theoremcantnflem3 7277* Lemma for cantnf 7279. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than  C has a normal form, we can use oeeu 6487 to factor  C into the form  ( ( A  ^o  X )  .o  Y )  +o  Z where  0  <  Y  <  A and  Z  <  ( A  ^o  X ) (and a fortiori  X  < 
B). Then since  Z  <  ( A  ^o  X )  <_ 
( A  ^o  X
)  .o  Y  <_  C,  Z has a normal form, and by appending the term  ( A  ^o  X )  .o  Y using cantnfp1 7267 we get a normal form for 
C. (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  C  e.  ( A  ^o  B ) )   &    |-  ( ph  ->  C  C_  ran  (  A CNF  B ) )   &    |-  ( ph  ->  (/)  e.  C )   &    |-  X  =  U. |^| { c  e.  On  |  C  e.  ( A  ^o  c ) }   &    |-  P  =  (
 iota d E. a  e.  On  E. b  e.  ( A  ^o  X ) ( d  = 
 <. a ,  b >.  /\  ( ( ( A 
 ^o  X )  .o  a )  +o  b
 )  =  C ) )   &    |-  Y  =  ( 1st `  P )   &    |-  Z  =  ( 2nd `  P )   &    |-  ( ph  ->  G  e.  S )   &    |-  ( ph  ->  ( ( A CNF  B ) `
  G )  =  Z )   &    |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `
  t ) ) )   =>    |-  ( ph  ->  C  e.  ran  (  A CNF  B ) )
 
Theoremcantnflem4 7278* Lemma for cantnf 7279. Complete the induction step of cantnflem3 7277. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  ( ph  ->  C  e.  ( A  ^o  B ) )   &    |-  ( ph  ->  C  C_  ran  (  A CNF  B ) )   &    |-  ( ph  ->  (/)  e.  C )   &    |-  X  =  U. |^| { c  e.  On  |  C  e.  ( A  ^o  c ) }   &    |-  P  =  (
 iota d E. a  e.  On  E. b  e.  ( A  ^o  X ) ( d  = 
 <. a ,  b >.  /\  ( ( ( A 
 ^o  X )  .o  a )  +o  b
 )  =  C ) )   &    |-  Y  =  ( 1st `  P )   &    |-  Z  =  ( 2nd `  P )   =>    |-  ( ph  ->  C  e.  ran  (  A CNF  B ) )
 
Theoremcantnf 7279* The Cantor Normal Form theorem. The function  ( A CNF  B ), which maps a finitely supported function from  B to  A to the sum  ( ( A  ^o  f ( a 1 ) )  o.  a 1 )  +o  ( ( A  ^o  f ( a 2 ) )  o.  a 2 )  +o 
... over all indexes  a  <  B such that  f ( a ) is nonzero, is an order isomorphism from the ordering  T of finitely supported functions to the set  ( A  ^o  B
) under the natural order. Setting 
A  =  om and letting  B be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 7263, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
 
Theoremoemapwe 7280* 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.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
 
Theoremcantnffval2 7281* An alternative definition of df-cnf 7247 which relies on cantnf 7279. (Note that although the use of  S seems self-referential, one can use cantnfdm 7249 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  T  =  { <. x ,  y >.  |  E. z  e.  B  (
 ( x `  z
 )  e.  ( y `
  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }   =>    |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S ) )
 
Theoremcantnff1o 7282 Simplify the isomorphism of cantnf 7279 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  (  A CNF  B )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  ( A CNF  B ) : S -1-1-onto-> ( A  ^o  B ) )
 
Theoremmapfien 7283* A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } ) )  e. 
 Fin }   &    |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } )
 )  e.  Fin }   &    |-  W  =  ( G `  Z )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ph  ->  G : B -1-1-onto-> D )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  D  e.  _V )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 f  e.  S  |->  ( G  o.  ( f  o.  F ) ) ) : S -1-1-onto-> T )
 
Theoremwemapwe 7284* 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.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  (
 ( x `  z
 ) S ( y `
  z )  /\  A. w  e.  A  ( z R w  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  U  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } )
 )  e.  Fin }   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  S  We  B )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  F  = OrdIso ( R ,  A )   &    |-  G  = OrdIso ( S ,  B )   &    |-  Z  =  ( G `
  (/) )   =>    |-  ( ph  ->  T  We  U )
 
Theoremoef1o 7285* A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption 
( F `  (/) )  =  (/) can be discharged using fveqf1o 5658.) (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  F : A -1-1-onto-> C )   &    |-  ( ph  ->  G : B -1-1-onto-> D )   &    |-  ( ph  ->  A  e.  ( On  \  1o ) )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  C  e.  On )   &    |-  ( ph  ->  D  e.  On )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  K  =  ( y  e.  { x  e.  ( A  ^m  B )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( F  o.  (
 y  o.  `' G ) ) )   &    |-  H  =  ( ( ( C CNF 
 D )  o.  K )  o.  `' ( A CNF 
 B ) )   =>    |-  ( ph  ->  H : ( A  ^o  B ) -1-1-onto-> ( C  ^o  D ) )
 
Theoremcnfcomlem 7286* Lemma for cnfcom 7287. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  ( ph  ->  I  e.  dom  G )   &    |-  ( ph  ->  O  e.  ( om  ^o  ( G `  I ) ) )   &    |-  ( ph  ->  ( T `  I ) : ( H `  I ) -1-1-onto-> O )   =>    |-  ( ph  ->  ( T `  suc  I ) : ( H `  suc  I ) -1-1-onto-> ( ( om  ^o  ( G `  I ) )  .o  ( F `
  ( G `  I ) ) ) )
 
Theoremcnfcom 7287* Any ordinal  B is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  ( ph  ->  I  e.  dom  G )   =>    |-  ( ph  ->  ( T `  suc  I ) : ( H `  suc  I
 )
 -1-1-onto-> ( ( om  ^o  ( G `  I ) )  .o  ( F `
  ( G `  I ) ) ) )
 
Theoremcnfcom2lem 7288* Lemma for cnfcom2 7289. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  (/)  e.  B )   =>    |-  ( ph  ->  dom  G  =  suc  U. dom  G )
 
Theoremcnfcom2 7289* Any nonzero ordinal  B is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  (/)  e.  B )   =>    |-  ( ph  ->  ( T `  dom  G ) : B -1-1-onto-> ( ( om  ^o  W )  .o  ( F `  W ) ) )
 
Theoremcnfcom3lem 7290* Lemma for cnfcom3 7291. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  om  C_  B )   =>    |-  ( ph  ->  W  e.  ( On  \  1o ) )
 
Theoremcnfcom3 7291* Any infinite ordinal  B is equinumerous to a power of  om. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 7293.) (Contributed by Mario Carneiro, 28-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  ( om  ^o  A ) )   &    |-  F  =  ( `' ( om CNF  A ) `  B )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  ( ph  ->  om  C_  B )   &    |-  X  =  ( u  e.  ( F `
  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( F `  W )  .o  v )  +o  u ) )   &    |-  Y  =  ( u  e.  ( F `  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( om  ^o  W )  .o  u )  +o  v ) )   &    |-  N  =  ( ( X  o.  `' Y )  o.  ( T `  dom  G ) )   =>    |-  ( ph  ->  N : B -1-1-onto-> ( om  ^o  W ) )
 
Theoremcnfcom3clem 7292* Lemma for cnfcom3c 7293. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  S  =  dom  ( om CNF  A )   &    |-  F  =  ( `' ( om CNF  A ) `  b )   &    |-  G  = OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )   &    |-  H  = seq𝜔 ( ( k  e. 
 _V ,  z  e. 
 _V  |->  ( M  +o  z ) ) ,  (/) )   &    |-  T  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  K ) ,  (/) )   &    |-  M  =  ( ( om  ^o  ( G `  k ) )  .o  ( F `  ( G `  k ) ) )   &    |-  K  =  ( ( x  e.  M  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( M  +o  x ) ) )   &    |-  W  =  ( G `  U. dom  G )   &    |-  X  =  ( u  e.  ( F `
  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( F `  W )  .o  v )  +o  u ) )   &    |-  Y  =  ( u  e.  ( F `  W ) ,  v  e.  ( om  ^o  W )  |->  ( ( ( om  ^o  W )  .o  u )  +o  v ) )   &    |-  N  =  ( ( X  o.  `' Y )  o.  ( T `  dom  G ) )   &    |-  L  =  ( b  e.  ( om  ^o  A )  |->  N )   =>    |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( g `  b ) : b -1-1-onto-> ( om  ^o  w ) ) )
 
Theoremcnfcom3c 7293* Wrap the construction of cnfcom3 7291 into an existence quantifier. For any  om  C_  b, there is a bijection from  b to some power of  om. Furthermore, this bijection is canonical , which means that we can find a single function 
g which will give such bijections for every  b less than some arbitrarily large bound  A. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( g `  b ) : b -1-1-onto-> ( om  ^o  w ) ) )
 
2.6.4  Transitive closure
 
Theoremtrcl 7294* For any set  A, show the properties of its transitive closure  C. 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 7295 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   &    |-  F  =  ( rec ( ( z  e.  _V  |->  ( z  u.  U. z
 ) ) ,  A )  |`  om )   &    |-  C  =  U_ y  e.  om  ( F `  y )   =>    |-  ( A  C_  C  /\  Tr  C  /\  A. x ( ( A  C_  x  /\  Tr  x ) 
 ->  C  C_  x )
 )
 
Theoremtz9.1 7295* Every set has a transitive closure (smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7294 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.)

 |-  A  e.  _V   =>    |-  E. x ( A  C_  x  /\  Tr  x  /\  A. y
 ( ( A  C_  y  /\  Tr  y ) 
 ->  x  C_  y ) )
 
Theoremtz9.1c 7296* Alternative expression for the existence of transitive closures tz9.1 7295: the intersection of all transitive sets containing  A is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   =>    |-  |^| { x  |  ( A  C_  x  /\  Tr  x ) }  e.  _V
 
Theoremepfrs 7297* The strong form of the Axiom of Regularity (no sethood requirement on  A), with the axiom itself present as an antecedent. See also zfregs 7298. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  ( (  _E  Fr  A  /\  A  =/=  (/) )  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfregs 7298* The strong form of the Axiom of Regularity, which does not require that  A be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 7297. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfregs2 7299* Alternate strong form of the Axiom of Regularity. Not every element of a non-empty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
 |-  ( A  =/=  (/)  ->  -.  A. x  e.  A  E. y
 ( y  e.  A  /\  y  e.  x ) )
 
Theoremen3lplem1 7300* Lemma for en3lp 7302. (Contributed by Alan Sare, 28-Oct-2011.)
 |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A ) 
 ->  ( x  =  A  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
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