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Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwecmpep 4401 The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)
 |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
 
Theoremwetrep 4402 An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
 |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
 
Theoremwefrc 4403* A non-empty (possibly proper) subclass of a class well-ordered by  _E has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by NM, 17-Feb-2004.)
 |-  ( (  _E  We  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
 
Theoremwe0 4404 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
 |-  R  We  (/)
 
Theoremwereu 4405* A subset of a well-ordered set has a unique minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( R  We  A  /\  ( B  e.  V  /\  B  C_  A  /\  B  =/=  (/) ) ) 
 ->  E! x  e.  B  A. y  e.  B  -.  y R x )
 
Theoremwereu2 4406* All nonempty (possibly proper) subclasses of  A, which has a well-founded relation  R, have  R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B 
 C_  A  /\  B  =/= 
 (/) ) )  ->  E! x  e.  B  A. y  e.  B  -.  y R x )
 
2.3.9  Ordinals
 
Syntaxword 4407 Extend the definition of a wff to include the ordinal predicate.
 wff  Ord  A
 
Syntaxcon0 4408 Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.)
 class  On
 
Syntaxwlim 4409 Extend the definition of a wff to include the limit ordinal predicate.
 wff  Lim  A
 
Syntaxcsuc 4410 Extend class notation to include the successor function.
 class  suc  A
 
Definitiondf-ord 4411 Define the ordinal predicate, which is true for a class that is transitive and is well-ordered by the epsilon relation. Variant of definition of [BellMachover] p. 468. (Contributed by NM, 17-Sep-1993.)
 |-  ( Ord  A  <->  ( Tr  A  /\  _E  We  A ) )
 
Definitiondf-on 4412 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
 |- 
 On  =  { x  |  Ord  x }
 
Definitiondf-lim 4413 Define the limit ordinal predicate, which is true for a non-empty ordinal that is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42. See dflim2 4464, dflim3 4654, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
 |-  ( Lim  A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
 
Definitiondf-suc 4414 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 6546). Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4483), so that the successor of any ordinal class is still an ordinal class (ordsuc 4621), simplifying certain proofs. Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.)
 |- 
 suc  A  =  ( A  u.  { A }
 )
 
Theoremordeq 4415 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
 
Theoremelong 4416 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  ( A  e.  On  <->  Ord  A ) )
 
Theoremelon 4417 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  On 
 <-> 
 Ord  A )
 
Theoremeloni 4418 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  On  ->  Ord  A )
 
Theoremelon2 4419 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
 |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
 
Theoremlimeq 4420 Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
 
Theoremordwe 4421 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 We  A )
 
Theoremordtr 4422 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  Tr  A )
 
Theoremordfr 4423 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 Fr  A )
 
Theoremordelss 4424 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  B  C_  A )
 
Theoremtrssord 4425 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
 |-  ( ( Tr  A  /\  A  C_  B  /\  Ord 
 B )  ->  Ord  A )
 
Theoremordirr 4426 Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.)
 |-  ( Ord  A  ->  -.  A  e.  A )
 
Theoremnordeq 4427 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  A  =/=  B )
 
Theoremordn2lp 4428 An ordinal class cannot an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  -.  ( A  e.  B  /\  B  e.  A ) )
 
Theoremtz7.5 4429* A subclass (possibly proper) of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.)
 |-  ( ( Ord  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
 
Theoremordelord 4430 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  Ord  B )
 
Theoremtron 4431 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
 |- 
 Tr  On
 
Theoremordelon 4432 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  B  e.  On )
 
Theoremonelon 4433 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
 
Theoremtz7.7 4434 Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)
 |-  ( ( Ord  A  /\  Tr  B )  ->  ( B  e.  A  <->  ( B  C_  A  /\  B  =/=  A ) ) )
 
Theoremordelssne 4435 Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B ) ) )
 
Theoremordelpss 4436 Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A 
 C.  B ) )
 
Theoremordsseleq 4437 For ordinal classes, subclass is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
 
Theoremordin 4438 The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
 
Theoremonin 4439 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B )  e.  On )
 
Theoremordtri3or 4440 A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
 )
 
Theoremordtri1 4441 A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
 
Theoremontri1 4442 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  -.  B  e.  A ) )
 
Theoremordtri2 4443 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
 
Theoremordtri3 4444 A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
 
Theoremordtri4 4445 A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )
 
Theoremorddisj 4446 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( Ord  A  ->  ( A  i^i  { A } )  =  (/) )
 
Theoremonfr 4447 The ordinal class is well-founded. This lemma is needed for ordon 4590 in order to eliminate the need for the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
 |- 
 _E  Fr  On
 
Theoremonelpss 4448 Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B 
 <->  ( A  C_  B  /\  A  =/=  B ) ) )
 
Theoremonsseleq 4449 Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  ( A  e.  B  \/  A  =  B ) ) )
 
Theoremonelss 4450 An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  e.  On  ->  ( B  e.  A  ->  B  C_  A )
 )
 
Theoremordtr1 4451 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
 |-  ( Ord  C  ->  ( ( A  e.  B  /\  B  e.  C ) 
 ->  A  e.  C ) )
 
Theoremordtr2 4452 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( ( Ord  A  /\  Ord  C )  ->  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  C ) )
 
Theoremordtr3 4453 Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.)
 |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  e.  B  ->  ( A  e.  C  \/  C  e.  B ) ) )
 
Theoremontr1 4454 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
 |-  ( C  e.  On  ->  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  C ) )
 
Theoremontr2 4455 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
 |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( ( A 
 C_  B  /\  B  e.  C )  ->  A  e.  C ) )
 
Theoremordunidif 4456 The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  U. ( A  \  B )  =  U. A )
 
Theoremordintdif 4457 If  B is smaller than  A, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
 |-  ( ( Ord  A  /\  Ord  B  /\  ( A  \  B )  =/=  (/) )  ->  B  =  |^| ( A  \  B ) )
 
Theoremonintss 4458* If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  On  ->  ( ps  ->  |^|
 { x  e.  On  |  ph }  C_  A ) )
 
Theoremoneqmini 4459* A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)
 |-  ( B  C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B ) )
 
Theoremord0 4460 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
 |- 
 Ord  (/)
 
Theorem0elon 4461 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
 |-  (/)  e.  On
 
Theoremord0eln0 4462 A non-empty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)
 |-  ( Ord  A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
 
Theoremon0eln0 4463 An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
 |-  ( A  e.  On  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
 
Theoremdflim2 4464 An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)
 |-  ( Lim  A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
 
Theoreminton 4465 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
 |- 
 |^| On  =  (/)
 
Theoremnlim0 4466 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |- 
 -.  Lim  (/)
 
Theoremlimord 4467 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
 |-  ( Lim  A  ->  Ord 
 A )
 
Theoremlimuni 4468 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
 |-  ( Lim  A  ->  A  =  U. A )
 
Theoremlimuni2 4469 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
 |-  ( Lim  A  ->  Lim  U. A )
 
Theorem0ellim 4470 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
 |-  ( Lim  A  ->  (/)  e.  A )
 
Theoremlimelon 4471 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
 |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
 
Theoremonn0 4472 The class of all ordinal numbers in not empty. (Contributed by NM, 17-Sep-1995.)
 |- 
 On  =/=  (/)
 
Theoremsuceq 4473 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =  B  ->  suc  A  =  suc  B )
 
Theoremelsuci 4474 Membership in a successor. This one-way implication does not require that either  A or  B be sets. (Contributed by NM, 6-Jun-1994.)
 |-  ( A  e.  suc  B 
 ->  ( A  e.  B  \/  A  =  B ) )
 
Theoremelsucg 4475 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
 |-  ( A  e.  V  ->  ( A  e.  suc  B  <-> 
 ( A  e.  B  \/  A  =  B ) ) )
 
Theoremelsuc2g 4476 Variant of membership in a successor, requiring that  B rather than  A be a set. (Contributed by NM, 28-Oct-2003.)
 |-  ( B  e.  V  ->  ( A  e.  suc  B  <-> 
 ( A  e.  B  \/  A  =  B ) ) )
 
Theoremelsuc 4477 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A  e.  suc 
 B 
 <->  ( A  e.  B  \/  A  =  B ) )
 
Theoremelsuc2 4478 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( B  e.  suc 
 A 
 <->  ( B  e.  A  \/  B  =  A ) )
 
Theoremnfsuc 4479 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
 |-  F/_ x A   =>    |-  F/_ x  suc  A
 
Theoremelelsuc 4480 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
 |-  ( A  e.  B  ->  A  e.  suc  B )
 
Theoremsucel 4481* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
 |-  ( suc  A  e.  B 
 <-> 
 E. x  e.  B  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
 
Theoremsuc0 4482 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
 |- 
 suc  (/)  =  { (/) }
 
Theoremsucprc 4483 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
 |-  ( -.  A  e.  _V 
 ->  suc  A  =  A )
 
Theoremunisuc 4484 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  U.
 suc  A  =  A )
 
Theoremsssucid 4485 A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
 |-  A  C_  suc  A
 
Theoremsucidg 4486 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
 |-  ( A  e.  V  ->  A  e.  suc  A )
 
Theoremsucid 4487 A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
Theoremnsuceq0 4488 No successor is empty. (Contributed by NM, 3-Apr-1995.)
 |- 
 suc  A  =/=  (/)
 
Theoremeqelsuc 4489 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( A  =  B  ->  A  e.  suc  B )
 
Theoremiunsuc 4490* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  U_ x  e.  suc  A B  =  ( U_ x  e.  A  B  u.  C )
 
Theoremsuctr 4491 The successor of a transtive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsuc 4492 A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
 
Theoremtrsuc2OLD 4493 Obsolete proof of suctr 4491 as of 5-Apr-2016. The successor of a transitive set is transitive. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsucss 4494 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
 |-  ( Tr  A  ->  ( B  e.  suc  A  ->  B  C_  A )
 )
 
Theoremordsssuc 4495 A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B )
 )
 
Theoremonsssuc 4496 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  A  e.  suc  B ) )
 
Theoremordsssuc2 4497 An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B )
 )
 
Theoremonmindif 4498 When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
 |-  ( ( A  C_  On  /\  B  e.  On )  ->  B  e.  |^| ( A  \  suc  B ) )
 
Theoremordnbtwn 4499 There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.)
 |-  ( Ord  A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
 
Theoremonnbtwn 4500 There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)
 |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
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