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Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheorembdcnulALT 9301 Alternate proof of bdcnul 9300. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 9279, or use the corresponding characterizations of its elements followed by bdelir 9282. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED  (/)
 
Theorembdeq0 9302 Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED  (/)
 
Theorembj-bd0el 9303 Boundedness of the formula "the empty set belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.)
BOUNDED  (/)
 
Theorembdcpw 9304 The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED    =>    BOUNDED  ~P
 
Theorembdcsn 9305 The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED  { }
 
Theorembdcpr 9306 The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED  { ,  }
 
Theorembdctp 9307 The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED  { ,  ,  }
 
Theorembdsnss 9308* Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED    =>    BOUNDED  { }  C_
 
Theorembdvsn 9309* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED  { }
 
Theorembdop 9310 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED 
 <. ,  >.
 
Theorembdcuni 9311 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
BOUNDED 
 U.
 
Theorembdcint 9312 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 
 |^|
 
Theorembdciun 9313* The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED    =>    BOUNDED  U_
 
Theorembdciin 9314* The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED    =>    BOUNDED  |^|_
 
Theorembdcsuc 9315 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 
 suc
 
Theorembdeqsuc 9316* Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
BOUNDED  suc
 
Theorembj-bdsucel 9317 Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.)
BOUNDED  suc
 
Theorembdcriota 9318* A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
BOUNDED    &       =>    BOUNDED  iota_
 
5.3.6  Bounded separation

In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.

 
Axiomax-bdsep 9319* Axiom scheme of bounded (or restricted, or Δ0) 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 3866. (Contributed by BJ, 5-Oct-2019.)
BOUNDED    =>     a b  b  a
 
Theorembdsep2 9320* Version of ax-bdsep 9319 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
BOUNDED    =>     b  b  a
 
Theorembdsepnft 9321* Closed form of bdsepnf 9322. Version of ax-bdsep 9319 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 19-Oct-2019.)
BOUNDED    =>     F/ b  b  b  a
 
Theorembdsepnf 9322* Version of ax-bdsep 9319 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 9323. (Contributed by BJ, 5-Oct-2019.)
 F/ b   &    BOUNDED    =>     b  b  a
 
TheorembdsepnfALT 9323* Alternate proof of bdsepnf 9322, not using bdsepnft 9321. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 F/ b   &    BOUNDED    =>     b  b  a
 
Theorembdzfauscl 9324* Closed form of the version of zfauscl 3868 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
BOUNDED    =>     V
 
Theorembdbm1.3ii 9325* Bounded version of bm1.3ii 3869. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
BOUNDED    &       =>   
 
Theorembj-nalset 9326* nalset 3878 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 
Theorembj-vprc 9327 vprc 3879 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 _V  _V
 
Theorembj-nvel 9328 nvel 3880 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 _V
 
Theorembj-vnex 9329 vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 _V
 
Theorembdinex1 9330 Bounded version of inex1 3882. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     _V   =>     i^i  _V
 
Theorembdinex2 9331 Bounded version of inex2 3883. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     _V   =>     i^i  _V
 
Theorembdinex1g 9332 Bounded version of inex1g 3884. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    =>     V  i^i  _V
 
Theorembdssex 9333 Bounded version of ssex 3885. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     _V   =>     C_  _V
 
Theorembdssexi 9334 Bounded version of ssexi 3886. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     _V   &     C_    =>     _V
 
Theorembdssexg 9335 Bounded version of ssexg 3887. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    =>     C_  C  _V
 
Theorembdssexd 9336 Bounded version of ssexd 3888. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 C   &     C_    &    BOUNDED    =>     _V
 
Theorembdrabexg 9337* Bounded version of rabexg 3891. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &    BOUNDED    =>     V  {  |  }  _V
 
Theorembj-inex 9338 The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 V  W  i^i  _V
 
Theorembj-intexr 9339 vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |^|  _V  =/=  (/)
 
Theorembj-intnexr 9340 vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |^|  _V 
 |^|  _V
 
Theorembj-zfpair2 9341 Proof of zfpair2 3936 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 { ,  }  _V
 
Theorembj-prexg 9342 Proof of prexg 3938 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 V  W  { ,  }  _V
 
Theorembj-snexg 9343 snexg 3927 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 V  { }  _V
 
Theorembj-snex 9344 snex 3928 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 _V   =>    
 { }  _V
 
Theorembj-sels 9345* If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
 V
 
Theorembj-axun2 9346* axun2 4138 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 
Theorembj-uniex2 9347* uniex2 4139 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 U.
 
Theorembj-uniex 9348 uniex 4140 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 _V   =>    
 U.  _V
 
Theorembj-uniexg 9349 uniexg 4141 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 V  U.  _V
 
Theorembj-unex 9350 unex 4142 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 _V   &     _V   =>     u.  _V
 
Theorembdunexb 9351 Bounded version of unexb 4143. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &    BOUNDED    =>     _V  _V  u. 
 _V
 
Theorembj-unexg 9352 unexg 4144 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 V  W  u.  _V
 
Theorembj-sucexg 9353 sucexg 4190 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 V  suc  _V
 
Theorembj-sucex 9354 sucex 4191 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 _V   =>    
 suc  _V
 
5.3.6.1  Delta_0-classical logic
 
Axiomax-bj-d0cl 9355 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED    =>    DECID
 
Theorembj-notbi 9356 Equivalence property for negation. TODO: minimize all theorems using notbid 591 and notbii 593. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
 
Theorembj-notbii 9357 Inference associated with bj-notbi 9356. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
   =>   
 
Theorembj-notbid 9358 Deduction form of bj-notbi 9356. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
   =>   
 
Theorembj-dcbi 9359 Equivalence property for DECID. TODO: solve conflict with dcbi 843; minimize dcbii 746 and dcbid 747 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
DECID DECID
 
Theorembj-d0clsepcl 9360 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID
 
5.3.6.2  Inductive classes and the class of natural numbers (finite ordinals)
 
Syntaxwind 9361 Syntax for inductive classes.
Ind
 
Definitiondf-bj-ind 9362* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind  (/)  suc
 
Theorembj-indsuc 9363 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
Ind  suc
 
Theorembj-indeq 9364 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
Ind Ind
 
Theorembj-bdind 9365 Boundedness of the formula "the setvar is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind
 
Theorembj-indint 9366* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind  |^| {  | Ind  }
 
Theorembj-dfom 9367 Alternate definition of  om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
 om  |^| {  | Ind  }
 
Theorembj-omind 9368  om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind  om
 
Theorembj-omssind 9369  om 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.)
 V Ind  om  C_
 
Theorembj-ssom 9370* A characterization of subclasses of  om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Ind  C_  C_  om
 
Theorembj-om 9371* A set is equal to  om if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
 V  om Ind Ind  C_
 
Theorembj-2inf 9372* Two formulations of the axiom of infinity (see ax-infvn 9375 and bj-omex 9376) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
 om  _V Ind Ind  C_
 
5.3.6.3  The first three Peano postulates

The first three Peano postulates do not require the axiom of infinity. We give constructive proofs (only the proof of the second postulate has to be modified). We also prove a preliminary version of the fifth Peano postulate.

 
Theorembj-peano2 9373 Constructive proof of peano2 4261. Temporary note: another possibility is to simply replace sucexg 4190 with bj-sucexg 9353 in the proof of peano2 4261. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 om  suc  om
 
Theorempeano5set 9374* Version of peano5 4264 when  om  i^i 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.)
 om  i^i  V  (/)  om  suc  om  C_
 
5.3.7  Axiom of infinity

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.

 
5.3.7.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 9375) and deduce that the class  om of finite ordinals is a set (bj-omex 9376).

 
Axiomax-infvn 9375* 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 4254) from which one then proves (omex 4259) using full separation that the wanted set exists. "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.)
Ind Ind  C_
 
Theorembj-omex 9376 Proof of omex 4259 from ax-infvn 9375. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
 om  _V
 
5.3.7.2  The remaining two Peano postulates

In this section, we give constructive proofs of the remaining two (the fourth and fifth) Peano postulates. More precisely, we prove from the core axioms of CZF that the set of finite ordinals satisfies the Peano postulates and thus provides a model for the set of natural numbers.

 
Theorembdpeano5 9377* Bounded version of peano5 4264. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    =>     (/)  om  suc  om  C_
 
Theoremspeano5 9378* Version of peano5 4264 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.)
 V  (/)  om  suc  om  C_
 
5.3.7.3  Bounded induction

In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction).

 
Theoremfindset 9379* Bounded induction (principle of induction when is assumed to be a set) allowing a proof from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See bdfind 9380 for a proof when is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 V  C_  om  (/)  suc  om
 
Theorembdfind 9380* Bounded induction (principle of induction when is assumed to be bounded), proved from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See findset 9379 for a proof when is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    =>     C_  om  (/)  suc  om
 
Theorembj-bdfindis 9381* 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 4266 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     F/   &     F/   &     F/   &     (/)    &       &     suc    =>     om  om
 
Theorembj-bdfindisg 9382* Version of bj-bdfindis 9381 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9381 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     F/   &     F/   &     F/   &     (/)    &       &     suc    &     F/_   &     F/   &       =>     om  om
 
Theorembj-bdfindes 9383 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 9381 for explanations. From this version, it is easy to prove the bounded version of findes 4269. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    =>     [. (/)  ].  om  [. suc  ].  om
 
Theorembj-nn0suc0 9384* Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9394. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 om  (/)  suc
 
Theorembj-nntrans 9385 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 om  C_
 
Theorembj-nntrans2 9386 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 om  Tr
 
Theorembj-nnelirr 9387 A natural number does not belong to itself. Version of elirr 4224 for natural numbers, which does not require ax-setind 4220. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
 om
 
Theorembj-nnen2lp 9388 A version of en2lp 4232 for natural numbers, which does not require ax-setind 4220.

Note: using this theorem and bj-nnelirr 9387, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

 om  om
 
Theorembj-peano4 9389 Remove from peano4 4263 dependency on ax-setind 4220. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
 om  om  suc  suc
 
Theorembj-omtrans 9390 The set  om is transitive. A natural number is included in  om.

The idea is to use bounded induction with the formula  C_ 
om. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with  C_  a and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

 om  C_ 
 om
 
Theorembj-omtrans2 9391 The set  om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 Tr  om
 
Theorembj-omord 9392 The set  om is an ordinal. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 Ord  om
 
Theorembj-omelon 9393 The set  om is an ordinal. Constructive proof of omelon 4274. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 om  On
 
Theorembj-nn0suc 9394* Proof of (biconditional form of) nn0suc 4270 from the core axioms of CZF. See also bj-nn0sucALT 9408. As a characterization of the elements of  om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 om  (/)  om  suc
 
5.3.8  Set induction

In this section, we add the axiom of set induction to the core axioms of CZF.

 
5.3.8.1  Set induction

In this section, we prove some variants of the axiom of set induction.

 
Theoremsetindft 9395* Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
 F/
 
Theoremsetindf 9396* Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
 F/   =>   
 
Theoremsetindis 9397* Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
 F/   &     F/   &     F/   &     F/   &       &       =>   
 
Axiomax-bdsetind 9398* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
BOUNDED    =>     a  a  a  a
 
Theorembdsetindis 9399* Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
BOUNDED    &     F/   &     F/   &     F/   &     F/   &       &       =>   
 
Theorembj-inf2vnlem1 9400* Lemma for bj-inf2vn 9404. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
 (/)  suc Ind
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