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Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfowdom 7301 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( ( F  e.  V  /\  F : Y -onto-> X )  ->  X  ~<_*  Y )
 
Theoremwdomref 7302 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  e.  V  ->  X  ~<_*  X )
 
Theorembrwdom2 7303* 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.)
 |-  ( Y  e.  V  ->  ( X  ~<_*  Y  <->  E. y  e.  ~P  Y E. z  z : y -onto-> X ) )
 
Theoremdomwdom 7304 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
 |-  ( X  ~<_  Y  ->  X  ~<_*  Y )
 
Theoremwdomtr 7305 Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( ( X  ~<_*  Y  /\  Y  ~<_*  Z )  ->  X  ~<_*  Z )
 
Theoremwdomen1 7306 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( A  ~<_*  C  <->  B  ~<_*  C ) )
 
Theoremwdomen2 7307 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~~  B  ->  ( C  ~<_*  A  <->  C  ~<_*  B ) )
 
Theoremwdompwdom 7308 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.)
 |-  ( X  ~<_*  Y  ->  ~P X  ~<_  ~P Y )
 
Theoremcanthwdom 7309 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7030, equivalent to canth 6310). (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 -.  ~P A  ~<_*  A
 
Theoremwdom2d 7310* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4147). (Contributed by Stefan O'Rear, 13-Feb-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  (
 ( ph  /\  x  e.  A )  ->  E. y  e.  B  x  =  X )   =>    |-  ( ph  ->  A  ~<_*  B )
 
Theoremwdomd 7311* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
 |-  ( ph  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  E. y  e.  B  x  =  X )   =>    |-  ( ph  ->  A  ~<_*  B )
 
Theorembrwdom3 7312* Condition for weak dominance with a condition reminiscent of wdomd 7311. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( X  ~<_*  Y  <->  E. f A. x  e.  X  E. y  e.  Y  x  =  ( f `  y ) ) )
 
Theorembrwdom3i 7313* Weak dominance implies existance of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
 |-  ( X  ~<_*  Y  ->  E. f A. x  e.  X  E. y  e.  Y  x  =  ( f `  y ) )
 
Theoremunwdomg 7314 Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( A  ~<_*  B  /\  C  ~<_*  D  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~<_*  ( B  u.  D ) )
 
Theoremxpwdomg 7315 Weak dominance of a cross product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( A  ~<_*  B  /\  C  ~<_*  D )  ->  ( A  X.  C )  ~<_*  ( B  X.  D ) )
 
Theoremwdomima2g 7316 A set is weakly dominant over its image under any function. This version of wdomimag 7317 is stated so as to avoid ax-rep 4147. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( Fun  F  /\  A  e.  V  /\  ( F " A )  e.  W )  ->  ( F " A )  ~<_*  A )
 
Theoremwdomimag 7317 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.)
 |-  ( ( Fun  F  /\  A  e.  V ) 
 ->  ( F " A ) 
 ~<_*  A )
 
Theoremunxpwdom2 7318 Lemma for unxpwdom 7319. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  X.  A )  ~~  ( B  u.  C )  ->  ( A  ~<_*  B  \/  A  ~<_  C ) )
 
Theoremunxpwdom 7319 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.)
 |-  ( ( A  X.  A )  ~<_  ( B  u.  C )  ->  ( A 
 ~<_*  B  \/  A  ~<_  C ) )
 
Theoremharwdom 7320 The Hartogs function is weakly dominated by  ~P ( X  X.  X
). This follows from a more precise analysis of the bound used in hartogs 7275 to prove that  (har `  X ) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( X  e.  V  ->  (har `  X )  ~<_*  ~P ( X  X.  X ) )
 
Theoremixpiunwdom 7321* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 6862 this shows that  U_ x  e.  A B and  X_ x  e.  A B have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( A  e.  V  /\  U_ x  e.  A  B  e.  W  /\  X_ x  e.  A  B  =/= 
 (/) )  ->  U_ x  e.  A  B  ~<_*  ( X_ x  e.  A  B  X.  A ) )
 
2.5  ZF Set Theory - add the Axiom of Regularity
 
2.5.1  Introduce the Axiom of Regularity
 
Axiomax-reg 7322* 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 7325) 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 7327). A stronger version that works for proper classes is proved as zfregs 7430. (Contributed by NM, 14-Aug-1993.)
 |-  ( E. y  y  e.  x  ->  E. y
 ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
 
Theoremaxreg2 7323* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
 |-  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremzfregcl 7324* The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A )
 
Theoremzfreg 7325* 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  A be a set, that can be proved with more difficulty (see zfregs 7430). (Contributed by NM, 26-Nov-1995.)
 |-  A  e.  _V   =>    |-  ( A  =/=  (/) 
 ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
 
Theoremzfreg2 7326* The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7325) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A  =/=  (/) 
 ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
 
Theoremelirrv 7327 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 7332 and efrirr 4390, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
 |- 
 -.  x  e.  x
 
Theoremelirr 7328 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.)
 |- 
 -.  A  e.  A
 
Theoremsucprcreg 7329 A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.)
 |-  ( -.  A  e.  _V  <->  suc 
 A  =  A )
 
Theoremruv 7330 The Russell class is equal to the universe  _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
 |- 
 { x  |  x  e/  x }  =  _V
 
TheoremruALT 7331 Alternate proof of Russell's Paradox ru 3003, simplified using (indirectly) the Axiom of Regularity ax-reg 7322. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 { x  |  x  e/  x }  e/  _V
 
Theoremzfregfr 7332 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
 |- 
 _E  Fr  A
 
Theoremen2lp 7333 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 7334 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 7335 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 7322 (via the preleq 7334 step). See df-op 3662 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 7336 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 7337* Assuming ax-reg 7322, 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 7338* 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 7355. (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 7339 Variation of Axiom of Infinity (using zfinf 7356 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 7340* Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7356 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 7341* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7342* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7343* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7344* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7345* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7346* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7347* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7325. (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 7348* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7349* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7350* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 7351* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 5767. (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 7352 Our Axiom of Infinity ax-inf 7355 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7340, 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 7357 and zfinf2 7359.) The main proof is provided by inf3lema 7341 through inf3lem7 7351, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7351. 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 7342.)
       F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 7343.)
       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 7345.)
       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 7346.)
       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 7347.)
       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 7348.)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 7349.)
       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 7350.)
       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 7351.)
       Q.E.D.
       
(Contributed by NM, 29-Oct-1996.)
 |- 
 E. x ( x  =/=  (/)  /\  x  C_  U. x )   =>    |- 
 om  e.  _V
 
Theoreminfeq5i 7353 Half of infeq5 7354. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( om  e.  _V  ->  E. x  x  C.  U. x )
 
Theoreminfeq5 7354 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 7360.) 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.)
 |-  ( 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 7355* 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 7339 and inf2 7340). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7359 and omex 7360 and are based on the (nontrivial) proof of inf3 7352. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7358. Theorem inf0 7338 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7362 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7358 requires this axiom along with Regularity ax-reg 7322 for its derivation (as theorem axinf2 7357 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7358 instead of this one. The derivation of this axiom from ax-inf2 7358 is shown by theorem axinf 7361.

Proofs should normally use the standard version ax-inf2 7358 instead of this axiom. (New usage is discouraged.) (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 7356* Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (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 7357* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 7355 and Regularity ax-reg 7322.

This theorem should not be referenced in any proof. Instead, use ax-inf2 7358 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 7358* 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 7359 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7357 above, using our version of Infinity ax-inf 7355 and the Axiom of Regularity ax-reg 7322. We will reference ax-inf2 7358 instead of axinf2 7357 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7355 from ax-inf2 7358 is shown by theorem axinf 7361. (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 7359* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7358 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 7360 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 7338.

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 4683 and  Fin  =  _V (the universe of all sets) by fineqv 7094. 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 4691 through peano5 4695 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

 |- 
 om  e.  _V
 
Theoremaxinf 7361* The first version of the Axiom of Infinity ax-inf 7355 proved from the second version ax-inf2 7358. Note that we didn't use ax-reg 7322, unlike the other direction axinf2 7357. (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 7362 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see theorem infeq5 7354). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)
 |- 
 E. x  x  C.  U. x
 
Theoremomelon 7363 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
 |- 
 om  e.  On
 
Theoremdfom3 7364* 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 7365* A simplification of elom 4675 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
 |-  ( A  e.  om  <->  A. x ( Lim  x  ->  A  e.  x ) )
 
Theoremdfom4 7366* A simplification of df-om 4673 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
 |- 
 om  =  { x  |  A. y ( Lim  y  ->  x  e.  y ) }
 
Theoremdfom5 7367  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 7368 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 7369 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 7370 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 7371 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 7372 The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |- 
 om  ~~  suc  om
 
Theoreminfdifsn 7373 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 7374 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 7375* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 7129 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 7376* Using the Axiom of Regularity in the form zfregfr 7332, 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 7377* Using the Axiom of Regularity in the form zfregfr 7332, 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 7378 Extend class notation with the Cantor normal form function.
 class CNF
 
Definitiondf-cnf 7379* 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 7413 of this function in terms of df-oi 7241. (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 7380* 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 7381* 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 7382* Lemma for cantnf 7411. The function appearing in cantnfval 7385 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 7383 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 7384 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 7385* 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 7386* 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 7387* 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 7388* 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 7389* 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 7390 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 7391 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 7392 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 7393* 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 7394* 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 7395* 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 7396* Lemma for cantnfp1 7399. (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 7397* Lemma for cantnfp1 7399. (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 7398* Lemma for cantnfp1 7399. (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 7399* 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 7400* The relation  T is a strict order on  S (a corollary of wemapso2 7283). (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 )
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