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Type | Label | Description |
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Statement | ||
Theorem | pwuninel2 6001 | The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | 2pwuninelg 6002 | The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.) |
Theorem | iunon 6003* | The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.) |
Syntax | wsmo 6004 | Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals. |
Definition | df-smo 6005* | Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | dfsmo2 6006* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.) |
Theorem | issmo 6007* | Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) |
Theorem | issmo2 6008* | Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smoeq 6009 | Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smodm 6010 | The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smores 6011 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Theorem | smores3 6012 | A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.) |
Theorem | smores2 6013 | A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Theorem | smodm2 6014 | The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | smofvon2dm 6015 | The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Theorem | iordsmo 6016 | The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Theorem | smo0 6017 | The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smofvon 6018 | If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smoel 6019 | If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiun 6020* | The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiso 6021 | If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.) |
Theorem | smoel2 6022 | A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Syntax | crecs 6023 | Notation for a function defined by strong transfinite recursion. |
recs | ||
Definition | df-recs 6024* |
Define a function recs on , the class of ordinal
numbers, by transfinite recursion given a rule which sets the next
value given all values so far. See df-irdg 6089 for more details on why
this definition is desirable. Unlike df-irdg 6089 which restricts the
update rule to use only the previous value, this version allows the
update rule to use all previous values, which is why it is
described
as "strong", although it is actually more primitive. See tfri1d 6054 and
tfri2d 6055 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | recseq 6025 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs recs | ||
Theorem | nfrecs 6026 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | tfrlem1 6027* | A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlem3ag 6028* | Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by Jim Kingdon, 5-Jul-2019.) |
Theorem | tfrlem3a 6029* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3 6030* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3-2d 6031* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlem4 6032* | Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem5 6033* | Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | recsfval 6034* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem6 6035* | Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem7 6036* | Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfrlem8 6037* | Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
recs | ||
Theorem | tfrlem9 6038* | Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.) |
recs recs recs | ||
Theorem | tfrfun 6039 | Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
recs | ||
Theorem | tfr2a 6040 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
recs | ||
Theorem | tfr0dm 6041 | Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.) |
recs | ||
Theorem | tfr0 6042 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
recs | ||
Theorem | tfrlemisucfn 6043* | We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlemisucaccv 6044* | We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibacc 6045* | Each element of is an acceptable function. Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibxssdm 6046* | The union of is defined on all ordinals. Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibfn 6047* | The union of is a function defined on . Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibex 6048* | The set exists. Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiubacc 6049* | The union of satisfies the recursion rule (lemma for tfrlemi1 6051). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiex 6050* | Lemma for tfrlemi1 6051. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi1 6051* |
We can define an acceptable function on any ordinal.
As with many of the transfinite recursion theorems, we have a hypothesis that states that is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi14d 6052* | The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
recs | ||
Theorem | tfrexlem 6053* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfri1d 6054* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere", which is stated here as . Alternately, would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2d 6055* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 6084). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfr1onlem3ag 6056* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3ag 6028 but for tfr1on 6069 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Theorem | tfr1onlem3 6057* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3 6030 but for tfr1on 6069 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.) |
Theorem | tfr1onlemssrecs 6058* | Lemma for tfr1on 6069. The union of functions acceptable for tfr1on 6069 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucfn 6059* | We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6069. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucaccv 6060* | Lemma for tfr1on 6069. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlembacc 6061* | Lemma for tfr1on 6069. Each element of is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembxssdm 6062* | Lemma for tfr1on 6069. The union of is defined on all elements of . (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembfn 6063* | Lemma for tfr1on 6069. The union of is a function defined on . (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlembex 6064* | Lemma for tfr1on 6069. The set exists. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlemubacc 6065* | Lemma for tfr1on 6069. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemex 6066* | Lemma for tfr1on 6069. (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemaccex 6067* |
We can define an acceptable function on any element of .
As with many of the transfinite recursion theorems, we have hypotheses that state that is a function and that it is defined up to . (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemres 6068* | Lemma for tfr1on 6069. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.) |
recs | ||
Theorem | tfr1on 6069* | Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfri1dALT 6070* |
Alternate proof of tfri1d 6054 in terms of tfr1on 6069.
Although this does show that the tfr1on 6069 proof is general enough to also prove tfri1d 6054, the tfri1d 6054 proof is simpler in places because it does not need to deal with being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.) |
recs | ||
Theorem | tfrcllemssrecs 6071* | Lemma for tfrcl 6083. The union of functions acceptable for tfrcl 6083 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucfn 6072* | We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6083. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucaccv 6073* | Lemma for tfrcl 6083. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllembacc 6074* | Lemma for tfrcl 6083. Each element of is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembxssdm 6075* | Lemma for tfrcl 6083. The union of is defined on all elements of . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembfn 6076* | Lemma for tfrcl 6083. The union of is a function defined on . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembex 6077* | Lemma for tfrcl 6083. The set exists. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemubacc 6078* | Lemma for tfrcl 6083. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemex 6079* | Lemma for tfrcl 6083. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemaccex 6080* |
We can define an acceptable function on any element of .
As with many of the transfinite recursion theorems, we have hypotheses that state that is a function and that it is defined up to . (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemres 6081* | Lemma for tfr1on 6069. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.) |
recs | ||
Theorem | tfrcldm 6082* | Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcl 6083* | Closure for transfinite recursion. As with tfr1on 6069, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfri1 6084* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere", which is stated here as . Alternately, would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2 6085* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 6084). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfri3 6086* | Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 6084). Finally, we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfrex 6087* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Syntax | crdg 6088 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 6089* |
Define a recursive definition generator on (the class of ordinal
numbers) with characteristic function and initial value .
This rather amazing operation allows us to define, with compact direct
definitions, functions that are usually defined in textbooks only with
indirect self-referencing recursive definitions. A recursive definition
requires advanced metalogic to justify - in particular, eliminating a
recursive definition is very difficult and often not even shown in
textbooks. On the other hand, the elimination of a direct definition is
a matter of simple mechanical substitution. The price paid is the
daunting complexity of our operation (especially when df-recs 6024
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple. In classical logic it would be easier to divide this definition
into cases based on whether the domain of is zero, a successor, or
a limit ordinal. Cases do not (in general) work that way in
intuitionistic logic, so instead we choose a definition which takes the
union of all the results of the characteristic function for ordinals in
the domain of .
This means that this definition has the expected
properties for increasing and continuous ordinal functions, which
include ordinal addition and multiplication.
For finite recursion we also define df-frec 6110 and for suitable characteristic functions df-frec 6110 yields the same result as restricted to , as seen at frecrdg 6127. Note: We introduce with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.) |
recs | ||
Theorem | rdgeq1 6090 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 6091 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 6092 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 6093* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 6094* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 6095* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 6096 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 6097 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 6098 | The recursive definition generator is a function on ordinal numbers. The condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6105; in cases like df-oadd 6139 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 6099* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 6100* | Value of the recursive definition generator. Lemma for rdgival 6101 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
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