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Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonpsssuc 4501 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
 |-  ( A  e.  On  ->  A  C.  suc  A )
 
Theoremordelsuc 4502 A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
 |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc 
 A  C_  B )
 )
 
Theoremonsucmin 4503* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
 |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x }
 )
 
Theoremordsucelsuc 4504 Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( Ord  B  ->  ( A  e.  B  <->  suc  A  e.  suc  B ) )
 
Theoremordsucsssuc 4505 The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  suc  A  C_  suc  B ) )
 
Theoremordsucuniel 4506 Given an element  A of the union of an ordinal  B,  suc  A is an element of  B itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)
 |-  ( Ord  B  ->  ( A  e.  U. B  <->  suc 
 A  e.  B ) )
 
Theoremordsucun 4507 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  suc  ( A  u.  B )  =  ( suc  A  u.  suc  B )
 )
 
Theoremordunpr 4508 The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C )  e.  { B ,  C } )
 
Theoremordunel 4509 The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  ( B  u.  C )  e.  A )
 
Theoremonsucuni 4510 A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)
 |-  ( A  C_  On  ->  A  C_  suc  U. A )
 
Theoremordsucuni 4511 An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)
 |-  ( Ord  A  ->  A 
 C_  suc  U. A )
 
Theoremorduniorsuc 4512 An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)
 |-  ( Ord  A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
 
Theoremunon 4513 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
 |- 
 U. On  =  On
 
Theoremordunisuc 4514 An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( Ord  A  ->  U.
 suc  A  =  A )
 
Theoremorduniss2 4515* The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)
 |-  ( Ord  A  ->  U.
 { x  e.  On  |  x  C_  A }  =  A )
 
Theoremonsucuni2 4516 A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
 
Theorem0elsuc 4517 The successor of an ordinal class contains the empty set. (Contributed by NM, 4-Apr-1995.)
 |-  ( Ord  A  ->  (/)  e. 
 suc  A )
 
Theoremlimon 4518 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
 |- 
 Lim  On
 
Theoremonssi 4519 An ordinal number is a subset of 
On. (Contributed by NM, 11-Aug-1994.)
 |-  A  e.  On   =>    |-  A  C_  On
 
Theoremonsuci 4520 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
 |-  A  e.  On   =>    |-  suc  A  e.  On
 
Theoremonuniorsuci 4521 An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.)
 |-  A  e.  On   =>    |-  ( A  =  U. A  \/  A  =  suc  U. A )
 
Theoremonuninsuci 4522* A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
 |-  A  e.  On   =>    |-  ( A  =  U. A  <->  -.  E. x  e. 
 On  A  =  suc  x )
 
Theoremonsucssi 4523 A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
 |-  A  e.  On   &    |-  B  e.  On   =>    |-  ( A  e.  B  <->  suc 
 A  C_  B )
 
Theoremnlimsucg 4524 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  -.  Lim  suc  A )
 
Theoremorduninsuc 4525* An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( Ord  A  ->  ( A  =  U. A  <->  -. 
 E. x  e.  On  A  =  suc  x ) )
 
Theoremordunisuc2 4526* An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
 |-  ( Ord  A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
 
Theoremordzsl 4527* An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
 |-  ( Ord  A  <->  ( A  =  (/) 
 \/  E. x  e.  On  A  =  suc  x  \/  Lim 
 A ) )
 
Theoremonzsl 4528* An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
 
Theoremdflim3 4529* An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( Lim  A  <->  ( Ord  A  /\  -.  ( A  =  (/) 
 \/  E. x  e.  On  A  =  suc  x ) ) )
 
Theoremdflim4 4530* An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A ) )
 
Theoremlimsuc 4531 The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
 |-  ( Lim  A  ->  ( B  e.  A  <->  suc  B  e.  A ) )
 
Theoremlimsssuc 4532 A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
 |-  ( Lim  A  ->  ( A  C_  B  <->  A  C_  suc  B ) )
 
Theoremnlimon 4533* Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.)
 |- 
 { x  e.  On  |  ( x  =  (/)  \/ 
 E. y  e.  On  x  =  suc  y ) }  =  { x  e.  On  |  -.  Lim  x }
 
Theoremlimuni3 4534* The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  Lim  x )  ->  Lim  U. A )
 
2.4.3  Transfinite induction
 
Theoremtfi 4535* The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if  A is a class of ordinal numbers with the property that every ordinal number included in  A also belongs to  A, then every ordinal number is in  A.

See theorem tfindes 4544 or tfinds 4541 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

 |-  ( ( A  C_  On  /\  A. x  e. 
 On  ( x  C_  A  ->  x  e.  A ) )  ->  A  =  On )
 
Theoremtfis 4536* Transfinite Induction Schema. If all ordinal numbers less than a given number  x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
 |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph )
 )   =>    |-  ( x  e.  On  -> 
 ph )
 
Theoremtfis2f 4537* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfis2 4538* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfis3 4539* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( A  e.  On  ->  ch )
 
Theoremtfisi 4540* A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  On )   &    |-  (
 ( ph  /\  ( R  e.  On  /\  R  C_  T )  /\  A. y ( S  e.  R  ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
 
Theoremtfinds 4541* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  On  ->  ( ch  ->  th )
 )   &    |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )   =>    |-  ( A  e.  On  ->  ta )
 
Theoremtfindsg 4542* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal  B instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  On  ->  ps )   &    |-  ( ( ( y  e.  On  /\  B  e.  On )  /\  B  C_  y )  ->  ( ch  ->  th )
 )   &    |-  ( ( ( Lim 
 x  /\  B  e.  On )  /\  B  C_  x )  ->  ( A. y  e.  x  ( B  C_  y  ->  ch )  -> 
 ph ) )   =>    |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  B  C_  A )  ->  ta )
 
Theoremtfindsg2 4543* Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal  suc  B instead of zero. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 5-Jan-2005.)
 |-  ( x  =  suc  B 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  On  ->  ps )   &    |-  ( ( y  e.  On  /\  B  e.  y )  ->  ( ch  ->  th ) )   &    |-  (
 ( Lim  x  /\  B  e.  x )  ->  ( A. y  e.  x  ( B  e.  y  ->  ch )  ->  ph )
 )   =>    |-  ( ( A  e.  On  /\  B  e.  A )  ->  ta )
 
Theoremtfindes 4544* Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
 |-  [. (/)  /  x ]. ph   &    |-  ( x  e.  On  ->  ( ph  ->  [. suc  x 
 /  x ]. ph )
 )   &    |-  ( Lim  y  ->  ( A. x  e.  y  ph 
 ->  [. y  /  x ].
 ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfinds2 4545* Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff  ta is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( ta  ->  ps )   &    |-  ( y  e. 
 On  ->  ( ta  ->  ( ch  ->  th )
 ) )   &    |-  ( Lim  x  ->  ( ta  ->  ( A. y  e.  x  ch  ->  ph ) ) )   =>    |-  ( x  e.  On  ->  ( ta  ->  ph )
 )
 
Theoremtfinds3 4546* Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( et  ->  ps )   &    |-  ( y  e. 
 On  ->  ( et  ->  ( ch  ->  th )
 ) )   &    |-  ( Lim  x  ->  ( et  ->  ( A. y  e.  x  ch  ->  ph ) ) )   =>    |-  ( A  e.  On  ->  ( et  ->  ta )
 )
 
2.4.4  The natural numbers (i.e. finite ordinals)
 
Syntaxcom 4547 Extend class notation to include the class of natural numbers.
 class  om
 
Definitiondf-om 4548* Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 4549 for an alternate definition. Later, when we assume the Axiom of Infinity, we show  om is a set in omex 7228, and  om can then be defined per dfom3 7232 (the smallest inductive set) and dfom4 7234.

Note: the natural numbers  om are a subset of the ordinal numbers df-on 4289. They are completely different from the natural numbers  NN (df-n 9627) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)

 |- 
 om  =  { x  e.  On  |  A. y
 ( Lim  y  ->  x  e.  y ) }
 
Theoremdfom2 4549 An alternate definition of the set of natural numbers  om. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the inner class builder of non-limit ordinal numbers (see nlimon 4533). (Contributed by NM, 1-Nov-2004.)
 |- 
 om  =  { x  e.  On  |  suc  x  C_ 
 { y  e.  On  |  -.  Lim  y } }
 
Theoremelom 4550* Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 7233. (Contributed by NM, 15-May-1994.)
 |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x ( Lim  x  ->  A  e.  x ) ) )
 
Theoremomsson 4551 Omega is a subset of  On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 om  C_  On
 
Theoremlimomss 4552 The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)
 |-  ( Lim  A  ->  om  C_  A )
 
Theoremnnon 4553 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theoremnnoni 4554 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  A  e.  om   =>    |-  A  e.  On
 
Theoremnnord 4555 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theoremordom 4556 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 Ord  om
 
Theoremelnn 4557 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
 |-  ( ( A  e.  B  /\  B  e.  om )  ->  A  e.  om )
 
Theoremomon 4558 The class of natural numbers  om is either an ordinal number (if we accept the Axiom of Infinity) or the proper class of all ordinal numbers (if we deny the Axiom of Infinity). Remark in [TakeutiZaring] p. 43. (Contributed by NM, 10-May-1998.)
 |-  ( om  e.  On  \/  om  =  On )
 
Theoremomelon2 4559 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
 |-  ( om  e.  _V  ->  om  e.  On )
 
Theoremnnlim 4560 A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
 |-  ( A  e.  om  ->  -.  Lim  A )
 
Theoremomssnlim 4561 The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 om  C_  { x  e. 
 On  |  -.  Lim  x }
 
Theoremlimom 4562 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
 |- 
 Lim  om
 
Theorempeano2b 4563 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
 |-  ( A  e.  om  <->  suc  A  e.  om )
 
Theoremnnsuc 4564* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( A  e.  om 
 /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
 
Theoremssnlim 4565* An ordinal subclass of non-limit ordinals is a class of natural numbers. Exercise 7 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.)
 |-  ( ( Ord  A  /\  A  C_  { x  e.  On  |  -.  Lim  x } )  ->  A  C_ 
 om )
 
2.4.5  Peano's postulates
 
Theorempeano1 4566 Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 4566 through peano5 4570 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.)
 |-  (/)  e.  om
 
Theorempeano2 4567 The successor of any natural number is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  om  ->  suc  A  e.  om )
 
Theorempeano3 4568 The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  om  ->  suc  A  =/=  (/) )
 
Theorempeano4 4569 Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theorempeano5 4570* The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction hypothesis, is derived from this theorem as theorem findes 4577. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A )
 
Theoremnn0suc 4571* A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/ 
 E. x  e.  om  A  =  suc  x ) )
 
2.4.6  Finite induction (for finite ordinals)
 
Theoremfind 4572* The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that  A is a set of natural numbers, zero belongs to 
A, and given any member of  A the member's successor also belongs to  A. The conclusion is that every natural number is in  A. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  om  /\  (/) 
 e.  A  /\  A. x  e.  A  suc  x  e.  A )   =>    |-  A  =  om
 
Theoremfinds 4573* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 14-Apr-1995.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  om  ->  ta )
 
Theoremfindsg 4574* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. The basis of this version is an arbitrary natural number  B instead of zero. (Contributed by NM, 16-Sep-1995.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  om  ->  ps )   &    |-  (
 ( ( y  e. 
 om  /\  B  e.  om )  /\  B  C_  y )  ->  ( ch 
 ->  th ) )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ta )
 
Theoremfinds2 4575* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( ta  ->  ps )   &    |-  ( y  e. 
 om  ->  ( ta  ->  ( ch  ->  th )
 ) )   =>    |-  ( x  e.  om  ->  ( ta  ->  ph )
 )
 
Theoremfinds1 4576* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( x  e.  om  -> 
 ph )
 
Theoremfindes 4577 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4544 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
 |-  [. (/)  /  x ]. ph   &    |-  ( x  e.  om  ->  (
 ph  ->  [. suc  x  /  x ]. ph ) )   =>    |-  ( x  e.  om  ->  ph )
 
2.4.7  Functions and relations
 
Syntaxcxp 4578 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4579 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4580 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4581 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4582 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4583 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4584 Extend the definition of a class to include the composition of two classes. (Read: The composition of  A and  B.)
 class  ( A  o.  B )
 
Syntaxwrel 4585 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Syntaxwfun 4586 Extend the definition of a wff to include the function predicate. (Read:  A is a function.)
 wff  Fun  A
 
Syntaxwfn 4587 Extend the definition of a wff to include the function predicate with a domain. (Read:  A is a function on  B.)
 wff  A  Fn  B
 
Syntaxwf 4588 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 
F maps  A into  B.)
 wff  F : A --> B
 
Syntaxwf1 4589 Extend the definition of a wff to include one-to-one functions. (Read:  F maps  A one-to-one into  B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
 wff  F : A -1-1-> B
 
Syntaxwfo 4590 Extend the definition of a wff to include onto functions. (Read:  F maps  A onto  B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
 wff  F : A -onto-> B
 
Syntaxwf1o 4591 Extend the definition of a wff to include one-to-one onto functions. (Read:  F maps  A one-to-one onto  B.) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.
 wff  F : A -1-1-onto-> B
 
Syntaxcfv 4592 Extend the definition of a class to include the value of a function. (Read: The value of  F at  A, or " F of  A.")
 class  ( F `  A )
 
Syntaxwiso 4593 Extend the definition of a wff to include the isomorphism property. (Read:  H is an  R,  S isomorphism of  A onto  B.)
 wff  H  Isom  R ,  S  ( A ,  B )
 
Definitiondf-xp 4594* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example,  ( { 1 ,  5 }  X.  {
2 ,  7 } )  =  ( { <. 1 ,  2 >. , 
<. 1 ,  7
>. }  u.  { <. 5 ,  2 >. , 
<. 5 ,  7
>. } ) (ex-xp 20636). Another example is that the set of rational numbers are defined in df-q 10196 using the cross-product  ( ZZ  X.  NN ); the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  X.  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
 
Definitiondf-rel 4595 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5031 and dfrel3 5037. (Contributed by NM, 1-Aug-1994.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-cnv 4596* Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if  A  e. 
_V and  B  e.  _V then  ( A `' R B  <-> 
B R A ), as proven in brcnv 4771 (see df-br 3921 and df-rel 4595 for more on relations). For example,  `' { <. 2 ,  6 >. , 
<. 3 ,  9
>. }  =  { <. 6 ,  2 >. , 
<. 9 ,  3
>. } (ex-cnv 20637). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
 |-  `' A  =  { <. x ,  y >.  |  y A x }
 
Definitiondf-co 4597* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example,  ( ( exp 
o.  cos ) `  0
)  =  _e (ex-co 20638) because  ( cos `  0 )  =  1 (see cos0 12304) and  ( exp `  1
)  =  _e (see df-e 12224). Note that Definition 7 of [Suppes] p. 63 reverses  A and  B, uses  /. instead of  o., and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)
 |-  ( A  o.  B )  =  { <. x ,  y >.  |  E. z
 ( x B z 
 /\  z A y ) }
 
Definitiondf-dm 4598* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  dom  F  =  { 2 ,  3 } (ex-dm 20639). Another example is the domain of the complex arctangent,  ( A  e. 
dom arctan 
<->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i ) ) (for proof see atandm 20004). Contrast with range (defined in df-rn 4599). For alternate definitions see dfdm2 5110, dfdm3 4774, and dfdm4 4779. The notation " dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
 |- 
 dom  A  =  { x  |  E. y  x A y }
 
Definitiondf-rn 4599 Define the range of a class. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ran  F  =  { 6 ,  9 } (ex-rn 20640). Contrast with domain (defined in df-dm 4598). For alternate definitions, see dfrn2 4775, dfrn3 4776, and dfrn4 5040. The notation " ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
 |- 
 ran  A  =  dom  `'  A
 
Definitiondf-res 4600 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression  ( exp  |`  RR ) (used in reeff1 12274) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 12223 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
>. } (ex-res 20641). (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V )
 )
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