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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvopab 3901* | Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.) |
Theorem | cbvopabv 3902* | Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.) |
Theorem | cbvopab1 3903* | Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Theorem | cbvopab2 3904* | Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.) |
Theorem | cbvopab1s 3905* | Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.) |
Theorem | cbvopab1v 3906* | Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
Theorem | cbvopab2v 3907* | Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.) |
Theorem | csbopabg 3908* | Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Theorem | unopab 3909 | Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.) |
Theorem | mpteq12f 3910 | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq12dva 3911* | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Theorem | mpteq12dv 3912* | An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq12 3913* | An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.) |
Theorem | mpteq1 3914* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq1d 3915* | An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.) |
Theorem | mpteq2ia 3916 | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq2i 3917 | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq12i 3918 | An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq2da 3919 | Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Theorem | mpteq2dva 3920* | Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) |
Theorem | mpteq2dv 3921* | An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.) |
Theorem | nfmpt 3922* | Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Theorem | nfmpt1 3923 | Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.) |
Theorem | cbvmptf 3924* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.) |
Theorem | cbvmpt 3925* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) |
Theorem | cbvmptv 3926* | Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) |
Theorem | mptv 3927* | Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
Syntax | wtr 3928 | Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35. |
Definition | df-tr 3929 | Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3930 (which is suggestive of the word "transitive"), dftr3 3932, dftr4 3933, and dftr5 3931. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.) |
Theorem | dftr2 3930* | An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.) |
Theorem | dftr5 3931* | An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.) |
Theorem | dftr3 3932* | An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.) |
Theorem | dftr4 3933 | An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.) |
Theorem | treq 3934 | Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.) |
Theorem | trel 3935 | In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | trel3 3936 | In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) |
Theorem | trss 3937 | An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) |
Theorem | trin 3938 | The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
Theorem | tr0 3939 | The empty set is transitive. (Contributed by NM, 16-Sep-1993.) |
Theorem | trv 3940 | The universe is transitive. (Contributed by NM, 14-Sep-2003.) |
Theorem | triun 3941* | The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | truni 3942* | The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.) |
Theorem | trint 3943* | The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) |
Theorem | trintssm 3944* | Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.) |
Theorem | trintssmOLD 3945* | Obsolete version of trintssm 3944 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Axiom | ax-coll 3946* | Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3999 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.) |
Theorem | repizf 3947* | Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 3946. It is identical to zfrep6 3948 except for the choice of a freeness hypothesis rather than a distinct variable constraint between and . (Contributed by Jim Kingdon, 23-Aug-2018.) |
Theorem | zfrep6 3948* | A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 3949 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.) |
Axiom | ax-sep 3949* |
The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p.
"Axioms of CZF and IZF" (with unnecessary quantifier removed,
and with a
condition replaced by a distinct
variable constraint between
and ).
The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with ) so that it asserts the existence of a collection only if it is smaller than some other collection that already exists. This prevents Russell's paradox ru 2837. In some texts, this scheme is called "Aussonderung" or the Subset Axiom. (Contributed by NM, 11-Sep-2006.) |
Theorem | axsep2 3950* | A less restrictive version of the Separation Scheme ax-sep 3949, where variables and can both appear free in the wff , which can therefore be thought of as . This version was derived from the more restrictive ax-sep 3949 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Theorem | zfauscl 3951* | Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3949, we invoke the Axiom of Extensionality (indirectly via vtocl 2673), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.) |
Theorem | bm1.3ii 3952* | Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3949. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.) |
Theorem | a9evsep 3953* | Derive a weakened version of ax-i9 1468, where and must be distinct, from Separation ax-sep 3949 and Extensionality ax-ext 2070. The theorem also holds (ax9vsep 3954), but in intuitionistic logic is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax9vsep 3954* | Derive a weakened version of ax-9 1469, where and must be distinct, from Separation ax-sep 3949 and Extensionality ax-ext 2070. In intuitionistic logic a9evsep 3953 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | zfnuleu 3955* | Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2073 to strengthen the hypothesis in the form of axnul 3956). (Contributed by NM, 22-Dec-2007.) |
Theorem | axnul 3956* |
The Null Set Axiom of ZF set theory: there exists a set with no
elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks,
this is presented as a separate axiom; here we show it can be derived
from Separation ax-sep 3949. This version of the Null Set Axiom tells us
that at least one empty set exists, but does not tell us that it is
unique - we need the Axiom of Extensionality to do that (see
zfnuleu 3955).
This theorem should not be referenced by any proof. Instead, use ax-nul 3957 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
Axiom | ax-nul 3957* | The Null Set Axiom of IZF set theory. It was derived as axnul 3956 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. Axiom 4 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by NM, 7-Aug-2003.) |
Theorem | 0ex 3958 | The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 3957. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | csbexga 3959 | The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
Theorem | csbexa 3960 | The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | nalset 3961* | No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) |
Theorem | vnex 3962 | The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
Theorem | vprc 3963 | The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.) |
Theorem | nvel 3964 | The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) |
Theorem | inex1 3965 | Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.) |
Theorem | inex2 3966 | Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.) |
Theorem | inex1g 3967 | Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.) |
Theorem | ssex 3968 | The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 3949 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
Theorem | ssexi 3969 | The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.) |
Theorem | ssexg 3970 | The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) |
Theorem | ssexd 3971 | A subclass of a set is a set. Deduction form of ssexg 3970. (Contributed by David Moews, 1-May-2017.) |
Theorem | difexg 3972 | Existence of a difference. (Contributed by NM, 26-May-1998.) |
Theorem | zfausab 3973* | Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.) |
Theorem | rabexg 3974* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) |
Theorem | rabex 3975* | Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.) |
Theorem | elssabg 3976* | Membership in a class abstraction involving a subset. Unlike elabg 2759, does not have to be a set. (Contributed by NM, 29-Aug-2006.) |
Theorem | inteximm 3977* | The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Theorem | intexr 3978 | If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Theorem | intnexr 3979 | If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Theorem | intexabim 3980 | The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Theorem | intexrabim 3981 | The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Theorem | iinexgm 3982* | The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by Jim Kingdon, 28-Aug-2018.) |
Theorem | inuni 3983* | The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.) |
Theorem | elpw2g 3984 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.) |
Theorem | elpw2 3985 | Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.) |
Theorem | pwnss 3986 | The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | pwne 3987 | No set equals its power set. The sethood antecedent is necessary; compare pwv 3647. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) |
Theorem | repizf2lem 3988 | Lemma for repizf2 3989. If we have a function-like proposition which provides at most one value of for each in a set , we can change "at most one" to "exactly one" by restricting the values of to those values for which the proposition provides a value of . (Contributed by Jim Kingdon, 7-Sep-2018.) |
Theorem | repizf2 3989* | Replacement. This version of replacement is stronger than repizf 3947 in the sense that does not need to map all values of in to a value of . The resulting set contains those elements for which there is a value of and in that sense, this theorem combines repizf 3947 with ax-sep 3949. Another variation would be but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.) |
Theorem | class2seteq 3990* | Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
Theorem | 0elpw 3991 | Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.) |
Theorem | 0nep0 3992 | The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
Theorem | 0inp0 3993 | Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.) |
Theorem | unidif0 3994 | The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) |
Theorem | iin0imm 3995* | An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Theorem | iin0r 3996* | If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Theorem | intv 3997 | The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.) |
Theorem | axpweq 3998* | Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4001 is not used by the proof. (Contributed by NM, 22-Jun-2009.) |
Theorem | bnd 3999* | A very strong generalization of the Axiom of Replacement (compare zfrep6 3948). Its strength lies in the rather profound fact that does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 3946. (Contributed by NM, 17-Oct-2004.) |
Theorem | bnd2 4000* | A variant of the Boundedness Axiom bnd 3999 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.) |
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