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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bdsnss 13101* | Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdvsn 13102* | Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdop 13103 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bdcuni 13104 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
BOUNDED | ||
Theorem | bdcint 13105 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdciun 13106* | The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdciin 13107* | The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcsuc 13108 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdeqsuc 13109* | 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 13110 | Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcriota 13111* | 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 13112* | 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 4046. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep1 13113* | Version of ax-bdsep 13112 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep2 13114* | Version of ax-bdsep 13112 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13113 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnft 13115* | Closed form of bdsepnf 13116. Version of ax-bdsep 13112 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 13113 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnf 13116* | Version of ax-bdsep 13112 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13117. Use bdsep1 13113 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnfALT 13117* | Alternate proof of bdsepnf 13116, not using bdsepnft 13115. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdzfauscl 13118* | Closed form of the version of zfauscl 4048 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
BOUNDED | ||
Theorem | bdbm1.3ii 13119* | Bounded version of bm1.3ii 4049. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-axemptylem 13120* | Lemma for bj-axempty 13121 and bj-axempty2 13122. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Theorem | bj-axempty 13121* | Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4053. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Theorem | bj-axempty2 13122* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 13121. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Theorem | bj-nalset 13123* | nalset 4058 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vprc 13124 | vprc 4060 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nvel 13125 | nvel 4061 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vnex 13126 | vnex 4059 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdinex1 13127 | Bounded version of inex1 4062. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex2 13128 | Bounded version of inex2 4063. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex1g 13129 | Bounded version of inex1g 4064. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssex 13130 | Bounded version of ssex 4065. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexi 13131 | Bounded version of ssexi 4066. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexg 13132 | Bounded version of ssexg 4067. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexd 13133 | Bounded version of ssexd 4068. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdrabexg 13134* | Bounded version of rabexg 4071. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bj-inex 13135 | The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intexr 13136 | intexr 4075 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intnexr 13137 | intnexr 4076 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-zfpair2 13138 | Proof of zfpair2 4132 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-prexg 13139 | Proof of prexg 4133 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-snexg 13140 | snexg 4108 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-snex 13141 | snex 4109 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-sels 13142* | 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 13143* | axun2 4357 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-uniex2 13144* | uniex2 4358 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-uniex 13145 | uniex 4359 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-uniexg 13146 | uniexg 4361 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-unex 13147 | unex 4362 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdunexb 13148 | Bounded version of unexb 4363. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bj-unexg 13149 | unexg 4364 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-sucexg 13150 | sucexg 4414 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-sucex 13151 | sucex 4415 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
Axiom | ax-bj-d0cl 13152 | Axiom for Δ_{0}-classical logic. (Contributed by BJ, 2-Jan-2020.) |
BOUNDED DECID | ||
Theorem | bj-d0clsepcl 13153 | Δ_{0}-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
DECID | ||
Syntax | wind 13154 | Syntax for inductive classes. |
Ind | ||
Definition | df-bj-ind 13155* | Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-indsuc 13156 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-indeq 13157 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
Ind Ind | ||
Theorem | bj-bdind 13158 | Boundedness of the formula "the setvar is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED Ind | ||
Theorem | bj-indint 13159* | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
Ind Ind | ||
Theorem | bj-indind 13160* | If is inductive and is "inductive in ", then is inductive. (Contributed by BJ, 25-Oct-2020.) |
Ind Ind | ||
Theorem | bj-dfom 13161 | 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 13162 | is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-omssind 13163 | 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 13164* | A characterization of subclasses of . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-om 13165* | 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 13166* | Two formulations of the axiom of infinity (see ax-infvn 13169 and bj-omex 13170) . (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 4508 and peano3 4510 already show this. In this section, we prove bj-peano2 13167 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms. | ||
Theorem | bj-peano2 13167 | Constructive proof of peano2 4509. Temporary note: another possibility is to simply replace sucexg 4414 with bj-sucexg 13150 in the proof of peano2 4509. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | peano5set 13168* | Version of peano5 4512 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 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 13169) and deduce that the class of finite ordinals is a set (bj-omex 13170). | ||
Axiom | ax-infvn 13169* | 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 4502) from which one then proves, using full separation, that the wanted set exists (omex 4507). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.) |
Ind Ind | ||
Theorem | bj-omex 13170 | Proof of omex 4507 from ax-infvn 13169. (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 13171* | Bounded version of peano5 4512. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | speano5 13172* | Version of peano5 4512 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 13173* | Bounded induction (principle of induction when is assumed to be a set) allowing a proof from basic constructive axioms. See find 4513 for a nonconstructive proof of the general case. See bdfind 13174 for a proof when is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdfind 13174* | Bounded induction (principle of induction when is assumed to be bounded), proved from basic constructive axioms. See find 4513 for a nonconstructive proof of the general case. See findset 13173 for a proof when is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindis 13175* | 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 4514 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4514, finds2 4515, finds1 4516. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindisg 13176* | Version of bj-bdfindis 13175 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13175 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindes 13177 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13175 for explanations. From this version, it is easy to prove the bounded version of findes 4517. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-nn0suc0 13178* | Constructive proof of a variant of nn0suc 4518. For a constructive proof of nn0suc 4518, see bj-nn0suc 13192. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans 13179 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans2 13180 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnelirr 13181 | A natural number does not belong to itself. Version of elirr 4456 for natural numbers, which does not require ax-setind 4452. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnen2lp 13182 |
A version of en2lp 4469 for natural numbers, which does not require
ax-setind 4452.
Note: using this theorem and bj-nnelirr 13181, one can remove dependency on ax-setind 4452 from nntri2 6390 and nndcel 6396; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-peano4 13183 | Remove from peano4 4511 dependency on ax-setind 4452. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-omtrans 13184 |
The set is
transitive. A natural number is included in
.
Constructive proof of elnn 4519.
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 13185 | The set is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnord 13186 | A natural number is an ordinal. Constructive proof of nnord 4525. Can also be proved from bj-nnelon 13187 if the latter is proved from bj-omssonALT 13191. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-nnelon 13187 | A natural number is an ordinal. Constructive proof of nnon 4523. Can also be proved from bj-omssonALT 13191. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-omord 13188 | The set is an ordinal. Constructive proof of ordom 4520. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omelon 13189 | The set is an ordinal. Constructive proof of omelon 4522. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omsson 13190 | Constructive proof of omsson 4526. See also bj-omssonALT 13191. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
Theorem | bj-omssonALT 13191 | Alternate proof of bj-omsson 13190. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | bj-nn0suc 13192* | Proof of (biconditional form of) nn0suc 4518 from the core axioms of CZF. See also bj-nn0sucALT 13206. 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 13193* | Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindf 13194* | Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindis 13195* | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
Axiom | ax-bdsetind 13196* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
BOUNDED | ||
Theorem | bdsetindis 13197* | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vnlem1 13198* | Lemma for bj-inf2vn 13202. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem2 13199* | Lemma for bj-inf2vnlem3 13200 and bj-inf2vnlem4 13201. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem3 13200* | Lemma for bj-inf2vn 13202. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED Ind |
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