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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | unisuc 5801 | 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.) |
Theorem | sssucid 5802 | 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.) |
Theorem | sucidg 5803 | Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
Theorem | sucid 5804 | 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.) |
Theorem | nsuceq0 5805 | No successor is empty. (Contributed by NM, 3-Apr-1995.) |
Theorem | eqelsuc 5806 | A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.) |
Theorem | iunsuc 5807* | Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Theorem | suctr 5808 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.) |
Theorem | suctrOLD 5809 | Obsolete proof of suctr 5808 as of 24-Sep-2021. (Contributed by Alan Sare, 11-Apr-2009.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | trsuc 5810 | 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.) |
Theorem | trsucss 5811 | A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
Theorem | ordsssuc 5812 | A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.) |
Theorem | onsssuc 5813 | A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.) |
Theorem | ordsssuc2 5814 | 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.) |
Theorem | onmindif 5815 | 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.) |
Theorem | ordnbtwn 5816 | 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.) (Proof shortened by JJ, 24-Sep-2021.) |
Theorem | ordnbtwnOLD 5817 | Obsolete proof of ordnbtwn 5816 as of 24-Sep-2021. (Contributed by NM, 21-Jun-1998.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | onnbtwn 5818 | There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.) |
Theorem | sucssel 5819 | A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
Theorem | orddif 5820 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | orduniss 5821 | An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.) |
Theorem | ordtri2or 5822 | A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | ordtri2or2 5823 | A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.) |
Theorem | ordtri2or3 5824 | A consequence of total ordering for ordinal classes. Similar to ordtri2or2 5823. (Contributed by David Moews, 1-May-2017.) |
Theorem | ordelinel 5825 | The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.) |
Theorem | ordelinelOLD 5826 | Obsolete proof of ordelinel 5825 as of 24-Sep-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | ordssun 5827 | Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.) |
Theorem | ordequn 5828 | The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.) |
Theorem | ordun 5829 | The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.) |
Theorem | ordunisssuc 5830 | A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.) |
Theorem | suc11 5831 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
Theorem | onordi 5832 | An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) |
Theorem | ontrci 5833 | An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) |
Theorem | onirri 5834 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | oneli 5835 | A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | onelssi 5836 | A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.) |
Theorem | onssneli 5837 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
Theorem | onssnel2i 5838 | An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) |
Theorem | onelini 5839 | An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
Theorem | oneluni 5840 | An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
Theorem | onunisuci 5841 | An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
Theorem | onsseli 5842 | Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.) |
Theorem | onun2i 5843 | The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) |
Theorem | unizlim 5844 | An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.) |
Theorem | on0eqel 5845 | An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.) |
Theorem | snsn0non 5846 | The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7069). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5847. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | onxpdisj 5847 | Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 5846. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | onnev 5848 | The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) |
Syntax | cio 5849 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 5850* | Soundness justification theorem for df-iota 5851. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 5851* |
Define Russell's definition description binder, which can be read as
"the unique such that ," where ordinarily contains
as a free
variable. Our definition is meaningful only when there
is exactly one
such that is
true (see iotaval 5862);
otherwise, it evaluates to the empty set (see iotanul 5866). Russell used
the inverted iota symbol to represent the binder.
Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 6624 (or iotacl 5874 for unbounded iota), as demonstrated in the proof of supub 8365. This can be easier than applying riotasbc 6626 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | dfiota2 5852* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 5853 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotad 5854 | Deduction version of nfiota 5855. (Contributed by NM, 18-Feb-2013.) |
Theorem | nfiota 5855 | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 5856 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 5857* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 5858 | Variable substitution in description binder. Compare sb8eu 2503. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 5859 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 5860 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 5861* | Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotaval 5862* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 5863 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iotaint 5864 | Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iota1 5865 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | iotanul 5866 | Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotassuni 5867 | The class is a subset of the union of all elements satisfying . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iotaex 5868 | Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iota4 5869 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 5870 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 5871* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 5872* | Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 5873 | Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 5874 |
Membership law for descriptions.
This can be useful for expanding an unbounded iota-based definition (see df-iota 5851). If you have a bounded iota-based definition, riotacl2 6624 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 5875 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 5876* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 5877* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 5878 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | dfiota4 5879 | The operation using the operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.) |
Theorem | dfiota4OLD 5880 | Obsolete proof of dfiota4 5879 as of 28-Oct-2021. (Contributed by Scott Fenton, 6-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | csbiota 5881* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.) |
Syntax | wfun 5882 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 5883 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 5884 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 5885 | Extend the definition of a wff to include one-to-one functions. (Read: maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. |
Syntax | wfo 5886 | Extend the definition of a wff to include onto functions. (Read: maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27. |
Syntax | wf1o 5887 | Extend the definition of a wff to include one-to-one onto functions. (Read: maps one-to-one onto .) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27. |
Syntax | cfv 5888 | Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .") |
Syntax | wiso 5889 | Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .) |
Definition | df-fun 5890 | Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true once we define cosine (df-cos 14801). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4729 with the maps-to notation (see df-mpt 4730 and df-mpt2 6655). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5891), a function with a given domain and codomain (df-f 5892), a one-to-one function (df-f1 5893), an onto function (df-fo 5894), or a one-to-one onto function (df-f1o 5895). For alternate definitions, see dffun2 5898, dffun3 5899, dffun4 5900, dffun5 5901, dffun6 5903, dffun7 5915, dffun8 5916, and dffun9 5917. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fn 5891 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 6047, dffn3 6054, dffn4 6121, and dffn5 6241. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f 5892 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. can be read as " is a function from to ". For alternate definitions, see dff2 6371, dff3 6372, and dff4 6373. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f1 5893 |
Define a one-to-one function. For equivalent definitions see dff12 6100
and dff13 6512. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We
use their notation ("1-1" above the arrow).
A one-to-one function is also called an "injection" or an "injective function", can be read as " is an injection from into ". Injections are precisely the monomorphisms in the category of sets and set functions, see setcmon 16737. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fo 5894 |
Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27.
We use their notation ("onto" under the arrow). For alternate
definitions, see dffo2 6119, dffo3 6374, dffo4 6375, and dffo5 6376.
An onto function is also called a "surjection" or a "surjective function", can be read as " is a surjection from onto ". Surjections are precisely the epimorphisms in the category of sets and set functions, see setcepi 16738. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f1o 5895 |
Define a one-to-one onto function. For equivalent definitions see
dff1o2 6142, dff1o3 6143, dff1o4 6145, and dff1o5 6146. Compare Definition
6.15(6) of [TakeutiZaring] p. 27.
We use their notation ("1-1" above
the arrow and "onto" below the arrow).
A one-to-one onto function is also called a "bijection" or a "bijective function", can be read as " is a bijection between and ". Bijections are precisely the isomorphisms in the category of sets and set functions, see setciso 16741. Therefore, two sets are called "isomorphic" if there is a bijection between them. According to isof1oidb 6574, two sets are isomorphic iff there is an isomorphism regarding the identity relation. In this case, the two sets are also "equinumerous", see bren 7964. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fv 5896* | Define the value of a function, , also known as function application. For example, (we prove this in cos0 14880 after we define cosine in df-cos 14801). Typically, function is defined using maps-to notation (see df-mpt 4730 and df-mpt2 6655), but this is not required. For example, (ex-fv 27300). Note that df-ov 6653 will define two-argument functions using ordered pairs as . This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6218 and fvprc 6185). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar notation for a function's value at , i.e. " of ," but without context-dependent notational ambiguity. Alternate definitions are dffv2 6271, dffv3 6187, fv2 6186, and fv3 6206 (the latter two previously required to be a set.) Restricted equivalents that require to be a function are shown in funfv 6265 and funfv2 6266. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6239. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6188. (Revised by Scott Fenton, 6-Oct-2017.) |
Definition | df-isom 5897* | Define the isomorphism predicate. We read this as " is an , isomorphism of onto ." Normally, and are ordering relations on and respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that and are subscripts. (Contributed by NM, 4-Mar-1997.) |
Theorem | dffun2 5898* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun3 5899* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun4 5900* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
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