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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bdeqsuc 10001* | Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bj-bdsucel 10002 | Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcriota 10003* | A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.) |
BOUNDED BOUNDED | ||
In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory. | ||
Axiom | ax-bdsep 10004* | Axiom scheme of bounded (or restricted, or Δ_{0}) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3875. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep1 10005* | Version of ax-bdsep 10004 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep2 10006* | Version of ax-bdsep 10004 with one DV condition removed and without initial universal quantifier. Use bdsep1 10005 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnft 10007* | Closed form of bdsepnf 10008. Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10005 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnf 10008* | Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10009. Use bdsep1 10005 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnfALT 10009* | Alternate proof of bdsepnf 10008, not using bdsepnft 10007. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdzfauscl 10010* | Closed form of the version of zfauscl 3877 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
BOUNDED | ||
Theorem | bdbm1.3ii 10011* | Bounded version of bm1.3ii 3878. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-axemptylem 10012* | Lemma for bj-axempty 10013 and bj-axempty2 10014. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.) |
Theorem | bj-axempty 10013* | Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3882. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.) |
Theorem | bj-axempty2 10014* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 10013. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.) |
Theorem | bj-nalset 10015* | nalset 3887 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vprc 10016 | vprc 3888 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nvel 10017 | nvel 3889 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vnex 10018 | vnex 3890 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdinex1 10019 | Bounded version of inex1 3891. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex2 10020 | Bounded version of inex2 3892. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex1g 10021 | Bounded version of inex1g 3893. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssex 10022 | Bounded version of ssex 3894. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexi 10023 | Bounded version of ssexi 3895. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexg 10024 | Bounded version of ssexg 3896. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexd 10025 | Bounded version of ssexd 3897. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdrabexg 10026* | Bounded version of rabexg 3900. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bj-inex 10027 | The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intexr 10028 | intexr 3904 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intnexr 10029 | intnexr 3905 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-zfpair2 10030 | Proof of zfpair2 3945 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-prexg 10031 | Proof of prexg 3947 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-snexg 10032 | snexg 3936 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-snex 10033 | snex 3937 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-sels 10034* | If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.) |
Theorem | bj-axun2 10035* | axun2 4172 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-uniex2 10036* | uniex2 4173 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-uniex 10037 | uniex 4174 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-uniexg 10038 | uniexg 4175 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-unex 10039 | unex 4176 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdunexb 10040 | Bounded version of unexb 4177. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bj-unexg 10041 | unexg 4178 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-sucexg 10042 | sucexg 4224 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-sucex 10043 | sucex 4225 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Axiom | ax-bj-d0cl 10044 | Axiom for Δ_{0}-classical logic. (Contributed by BJ, 2-Jan-2020.) |
BOUNDED DECID | ||
Theorem | bj-notbi 10045 | Equivalence property for negation. TODO: minimize all theorems using notbid 592 and notbii 594. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
Theorem | bj-notbii 10046 | Inference associated with bj-notbi 10045. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
Theorem | bj-notbid 10047 | Deduction form of bj-notbi 10045. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
Theorem | bj-dcbi 10048 | Equivalence property for DECID. TODO: solve conflict with dcbi 844; minimize dcbii 747 and dcbid 748 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
DECID DECID | ||
Theorem | bj-d0clsepcl 10049 | Δ_{0}-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
DECID | ||
Syntax | wind 10050 | Syntax for inductive classes. |
Ind | ||
Definition | df-bj-ind 10051* | Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-indsuc 10052 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-indeq 10053 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
Ind Ind | ||
Theorem | bj-bdind 10054 | Boundedness of the formula "the setvar is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED Ind | ||
Theorem | bj-indint 10055* | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
Ind Ind | ||
Theorem | bj-indind 10056* | If is inductive and is "inductive in ", then is inductive. (Contributed by BJ, 25-Oct-2020.) |
Ind Ind | ||
Theorem | bj-dfom 10057 | Alternate definition of , as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-omind 10058 | is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-omssind 10059 | is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-ssom 10060* | A characterization of subclasses of . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-om 10061* | A set is equal to if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind Ind | ||
Theorem | bj-2inf 10062* | Two formulations of the axiom of infinity (see ax-infvn 10066 and bj-omex 10067) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind Ind | ||
The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4317 and peano3 4319 already show this. In this section, we prove bj-peano2 10063 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms. | ||
Theorem | bj-peano2 10063 | Constructive proof of peano2 4318. Temporary note: another possibility is to simply replace sucexg 4224 with bj-sucexg 10042 in the proof of peano2 4318. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | peano5set 10064* | Version of peano5 4321 when is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | peano5setOLD 10065* | Obsolete version of peano5set 10064 as of 26-Oct-2020. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
In the absence of full separation, the axiom of infinity has to be stated more precisely, as the existence of the smallest class containing the empty set and the successor of each of its elements. | ||
In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 10066) and deduce that the class of finite ordinals is a set (bj-omex 10067). | ||
Axiom | ax-infvn 10066* | Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4311) from which one then proves, using full separation, that the wanted set exists (omex 4316). "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.) |
Ind Ind | ||
Theorem | bj-omex 10067 | Proof of omex 4316 from ax-infvn 10066. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
In this section, we give constructive proofs of two versions of Peano's fifth postulate. | ||
Theorem | bdpeano5 10068* | Bounded version of peano5 4321. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | speano5 10069* | Version of peano5 4321 when is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction). We also prove Peano's fourth postulate. Together with the results from the previous sections, this proves from the core axioms of CZF (with infinity) that the set of finite ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers. | ||
Theorem | findset 10070* | Bounded induction (principle of induction when is assumed to be a set) allowing a proof from basic constructive axioms. See find 4322 for a nonconstructive proof of the general case. See bdfind 10071 for a proof when is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdfind 10071* | Bounded induction (principle of induction when is assumed to be bounded), proved from basic constructive axioms. See find 4322 for a nonconstructive proof of the general case. See findset 10070 for a proof when is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindis 10072* | Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4323 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4323, finds2 4324, finds1 4325. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindisg 10073* | Version of bj-bdfindis 10072 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 10072 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindes 10074 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 10072 for explanations. From this version, it is easy to prove the bounded version of findes 4326. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-nn0suc0 10075* | Constructive proof of a variant of nn0suc 4327. For a constructive proof of nn0suc 4327, see bj-nn0suc 10089. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans 10076 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans2 10077 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnelirr 10078 | A natural number does not belong to itself. Version of elirr 4266 for natural numbers, which does not require ax-setind 4262. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnen2lp 10079 |
A version of en2lp 4278 for natural numbers, which does not require
ax-setind 4262.
Note: using this theorem and bj-nnelirr 10078, one can remove dependency on ax-setind 4262 from nntri2 6073 and nndcel 6078; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-peano4 10080 | Remove from peano4 4320 dependency on ax-setind 4262. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-omtrans 10081 |
The set is
transitive. A natural number is included in
.
Constructive proof of elnn 4328.
The idea is to use bounded induction with the formula . This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omtrans2 10082 | The set is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnord 10083 | A natural number is an ordinal. Constructive proof of nnord 4334. Can also be proved from bj-nnelon 10084 if the latter is proved from bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-nnelon 10084 | A natural number is an ordinal. Constructive proof of nnon 4332. Can also be proved from bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-omord 10085 | The set is an ordinal. Constructive proof of ordom 4329. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omelon 10086 | The set is an ordinal. Constructive proof of omelon 4331. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omsson 10087 | Constructive proof of omsson 4335. See also bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
Theorem | bj-omssonALT 10088 | Alternate proof of bj-omsson 10087. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | bj-nn0suc 10089* | Proof of (biconditional form of) nn0suc 4327 from the core axioms of CZF. See also bj-nn0sucALT 10103. As a characterization of the elements of , this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
In this section, we add the axiom of set induction to the core axioms of CZF. | ||
In this section, we prove some variants of the axiom of set induction. | ||
Theorem | setindft 10090* | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindf 10091* | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindis 10092* | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
Axiom | ax-bdsetind 10093* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
BOUNDED | ||
Theorem | bdsetindis 10094* | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vnlem1 10095* | Lemma for bj-inf2vn 10099. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem2 10096* | Lemma for bj-inf2vnlem3 10097 and bj-inf2vnlem4 10098. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem3 10097* | Lemma for bj-inf2vn 10099. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED Ind | ||
Theorem | bj-inf2vnlem4 10098* | Lemma for bj-inf2vn2 10100. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vn 10099* | A sufficient condition for to be a set. See bj-inf2vn2 10100 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vn2 10100* | A sufficient condition for to be a set; unbounded version of bj-inf2vn 10099. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
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