HomeHome Intuitionistic Logic Explorer
Theorem List (p. 132 of 133)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdsnss 13101* Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  { x }  C_  A
 
Theorembdvsn 13102* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  x  =  { y }
 
Theorembdop 13103 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
 |- BOUNDED 
 <. x ,  y >.
 
Theorembdcuni 13104 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
 |- BOUNDED 
 U. x
 
Theorembdcint 13105 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED 
 |^| x
 
Theorembdciun 13106* The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  U_ x  e.  y  A
 
Theorembdciin 13107* The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  |^|_ x  e.  y  A
 
Theorembdcsuc 13108 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED 
 suc  x
 
Theorembdeqsuc 13109* Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
 |- BOUNDED  x  =  suc  y
 
Theorembj-bdsucel 13110 Boundedness of the formula "the successor of the setvar  x belongs to the setvar  y". (Contributed by BJ, 30-Nov-2019.)
 |- BOUNDED  suc  x  e.  y
 
Theorembdcriota 13111* A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
 |- BOUNDED  ph   &    |-  E! x  e.  y  ph   =>    |- BOUNDED  ( iota_ x  e.  y  ph )
 
11.2.8  CZF: Bounded separation

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

 
Axiomax-bdsep 13112* 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 4046. (Contributed by BJ, 5-Oct-2019.)
 |- BOUNDED  ph   =>    |- 
 A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
 )
 
Theorembdsep1 13113* Version of ax-bdsep 13112 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
 |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
Theorembdsep2 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  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
Theorembdsepnft 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  ph   =>    |-  ( A. x F/ b ph  ->  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
 ) )
 
Theorembdsepnf 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.)
 |-  F/ b ph   &    |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
TheorembdsepnfALT 13117* Alternate proof of bdsepnf 13116, not using bdsepnft 13115. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F/ b ph   &    |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
Theorembdzfauscl 13118* Closed form of the version of zfauscl 4048 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
 |- BOUNDED  ph   =>    |-  ( A  e.  V  ->  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph ) ) )
 
Theorembdbm1.3ii 13119* Bounded version of bm1.3ii 4049. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |- BOUNDED  ph   &    |-  E. x A. y ( ph  ->  y  e.  x )   =>    |-  E. x A. y ( y  e.  x  <->  ph )
 
Theorembj-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.)
 |-  E. x A. y ( y  e.  x  -> F.  )
 
Theorembj-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.)
 |-  E. x A. y  e.  x F.
 
Theorembj-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.)
 |-  E. x A. y  -.  y  e.  x
 
Theorembj-nalset 13123* nalset 4058 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  E. x A. y  y  e.  x
 
Theorembj-vprc 13124 vprc 4060 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  _V  e.  _V
 
Theorembj-nvel 13125 nvel 4061 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  _V  e.  A
 
Theorembj-vnex 13126 vnex 4059 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  E. x  x  =  _V
 
Theorembdinex1 13127 Bounded version of inex1 4062. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   &    |-  A  e.  _V   =>    |-  ( A  i^i  B )  e. 
 _V
 
Theorembdinex2 13128 Bounded version of inex2 4063. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   &    |-  A  e.  _V   =>    |-  ( B  i^i  A )  e. 
 _V
 
Theorembdinex1g 13129 Bounded version of inex1g 4064. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   =>    |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theorembdssex 13130 Bounded version of ssex 4065. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theorembdssexi 13131 Bounded version of ssexi 4066. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |-  B  e.  _V   &    |-  A  C_  B   =>    |-  A  e.  _V
 
Theorembdssexg 13132 Bounded version of ssexg 4067. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   =>    |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
 
Theorembdssexd 13133 Bounded version of ssexd 4068. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A  C_  B )   &    |- BOUNDED  A   =>    |-  ( ph  ->  A  e.  _V )
 
Theorembdrabexg 13134* Bounded version of rabexg 4071. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  ph   &    |- BOUNDED  A   =>    |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theorembj-inex 13135 The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A  i^i  B )  e.  _V )
 
Theorembj-intexr 13136 intexr 4075 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
 
Theorembj-intnexr 13137 intnexr 4076 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( |^| A  =  _V  ->  -. 
 |^| A  e.  _V )
 
Theorembj-zfpair2 13138 Proof of zfpair2 4132 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  { x ,  y }  e.  _V
 
Theorembj-prexg 13139 Proof of prexg 4133 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B }  e.  _V )
 
Theorembj-snexg 13140 snexg 4108 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { A }  e.  _V )
 
Theorembj-snex 13141 snex 4109 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 { A }  e.  _V
 
Theorembj-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.)
 |-  ( A  e.  V  ->  E. x  A  e.  x )
 
Theorembj-axun2 13143* axun2 4357 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y A. z ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
 
Theorembj-uniex2 13144* uniex2 4358 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y  y  =  U. x
 
Theorembj-uniex 13145 uniex 4359 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 U. A  e.  _V
 
Theorembj-uniexg 13146 uniexg 4361 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  U. A  e.  _V )
 
Theorembj-unex 13147 unex 4362 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  u.  B )  e. 
 _V
 
Theorembdunexb 13148 Bounded version of unexb 4363. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |- BOUNDED  B   =>    |-  ( ( A  e.  _V 
 /\  B  e.  _V ) 
 <->  ( A  u.  B )  e.  _V )
 
Theorembj-unexg 13149 unexg 4364 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A  u.  B )  e.  _V )
 
Theorembj-sucexg 13150 sucexg 4414 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  suc 
 A  e.  _V )
 
Theorembj-sucex 13151 sucex 4415 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 suc  A  e.  _V
 
11.2.8.1  Delta_0-classical logic
 
Axiomax-bj-d0cl 13152 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
 |- BOUNDED  ph   =>    |- DECID  ph
 
Theorembj-d0clsepcl 13153 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
 |- DECID  ph
 
11.2.8.2  Inductive classes and the class of natural numbers (finite ordinals)
 
Syntaxwind 13154 Syntax for inductive classes.
 wff Ind  A
 
Definitiondf-bj-ind 13155* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
 |-  (Ind  A 
 <->  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A ) )
 
Theorembj-indsuc 13156 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
 |-  (Ind  A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
 
Theorembj-indeq 13157 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
 |-  ( A  =  B  ->  (Ind 
 A 
 <-> Ind 
 B ) )
 
Theorembj-bdind 13158 Boundedness of the formula "the setvar  x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
 |- BOUNDED Ind  x
 
Theorembj-indint 13159* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
 |- Ind  |^| { x  e.  A  | Ind  x }
 
Theorembj-indind 13160* If  A is inductive and  B is "inductive in  A", then  ( A  i^i  B ) is inductive. (Contributed by BJ, 25-Oct-2020.)
 |-  (
 (Ind  A  /\  ( (/)  e.  B  /\  A. x  e.  A  ( x  e.  B  ->  suc  x  e.  B ) ) ) 
 -> Ind  ( A  i^i  B ) )
 
Theorembj-dfom 13161 Alternate definition of  om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
 |-  om  =  |^| { x  | Ind  x }
 
Theorembj-omind 13162  om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
 |- Ind  om
 
Theorembj-omssind 13163  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.)
 |-  ( A  e.  V  ->  (Ind 
 A  ->  om  C_  A ) )
 
Theorembj-ssom 13164* A characterization of subclasses of  om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A. x (Ind  x  ->  A  C_  x )  <->  A  C_  om )
 
Theorembj-om 13165* 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.)
 |-  ( A  e.  V  ->  ( A  =  om  <->  (Ind  A  /\  A. x (Ind  x  ->  A  C_  x ) ) ) )
 
Theorembj-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.)
 |-  ( om  e.  _V  <->  E. x (Ind  x  /\  A. y (Ind  y  ->  x  C_  y )
 ) )
 
11.2.8.3  The first three Peano postulates

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.

 
Theorembj-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.)
 |-  ( A  e.  om  ->  suc  A  e.  om )
 
Theorempeano5set 13168* Version of peano5 4512 when  om  i^i  A 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  A )  e.  V  ->  (
 ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc 
 x  e.  A ) )  ->  om  C_  A ) )
 
11.2.9  CZF: 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.

 
11.2.9.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13169) and deduce that the class  om of finite ordinals is a set (bj-omex 13170).

 
Axiomax-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.)
 |-  E. x (Ind  x  /\  A. y
 (Ind  y  ->  x  C_  y ) )
 
Theorembj-omex 13170 Proof of omex 4507 from ax-infvn 13169. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
 |-  om  e.  _V
 
11.2.9.2  Peano's fifth postulate

In this section, we give constructive proofs of two versions of Peano's fifth postulate.

 
Theorembdpeano5 13171* Bounded version of peano5 4512. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   =>    |-  ( ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A )
 
Theoremspeano5 13172* Version of peano5 4512 when  A 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.)
 |-  (
 ( A  e.  V  /\  (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc 
 x  e.  A ) )  ->  om  C_  A )
 
11.2.9.3  Bounded induction and Peano's fourth postulate

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.

 
Theoremfindset 13173* Bounded induction (principle of induction when  A 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  A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  ( ( A  C_  om  /\  (/) 
 e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
 )
 
Theorembdfind 13174* Bounded induction (principle of induction when  A 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  A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   =>    |-  ( ( A  C_  om 
 /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
 
Theorembj-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  ph   &    |-  F/ x ps   &    |-  F/ x ch   &    |-  F/ x th   &    |-  ( x  =  (/)  ->  ( ps  ->  ph ) )   &    |-  ( x  =  y  ->  (
 ph  ->  ch ) )   &    |-  ( x  =  suc  y  ->  ( th  ->  ph ) )   =>    |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
 
Theorembj-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  ph   &    |-  F/ x ps   &    |-  F/ x ch   &    |-  F/ x th   &    |-  ( x  =  (/)  ->  ( ps  ->  ph ) )   &    |-  ( x  =  y  ->  (
 ph  ->  ch ) )   &    |-  ( x  =  suc  y  ->  ( th  ->  ph ) )   &    |-  F/_ x A   &    |-  F/ x ta   &    |-  ( x  =  A  ->  (
 ph  ->  ta ) )   =>    |-  ( ( ps 
 /\  A. y  e.  om  ( ch  ->  th )
 )  ->  ( A  e.  om  ->  ta )
 )
 
Theorembj-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  ph   =>    |-  ( ( [. (/)  /  x ].
 ph  /\  A. x  e. 
 om  ( ph  ->  [.
 suc  x  /  x ].
 ph ) )  ->  A. x  e.  om  ph )
 
Theorembj-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.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
 
Theorembj-nntrans 13179 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  ( B  e.  A  ->  B 
 C_  A ) )
 
Theorembj-nntrans2 13180 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  Tr  A )
 
Theorembj-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.)
 |-  ( A  e.  om  ->  -.  A  e.  A )
 
Theorembj-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.)

 |-  (
 ( A  e.  om  /\  B  e.  om )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
 
Theorembj-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.)
 |-  (
 ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theorembj-omtrans 13184 The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4519.

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

 |-  ( A  e.  om  ->  A  C_ 
 om )
 
Theorembj-omtrans2 13185 The set  om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  Tr  om
 
Theorembj-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.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theorembj-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.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theorembj-omord 13188 The set  om is an ordinal. Constructive proof of ordom 4520. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  Ord  om
 
Theorembj-omelon 13189 The set  om is an ordinal. Constructive proof of omelon 4522. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  om  e.  On
 
Theorembj-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.
 |-  om  C_ 
 On
 
Theorembj-omssonALT 13191 Alternate proof of bj-omsson 13190. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  om  C_ 
 On
 
Theorembj-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  om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  <->  ( A  =  (/) 
 \/  E. x  e.  om  A  =  suc  x ) )
 
11.2.10  CZF: Set induction

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

 
11.2.10.1  Set induction

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

 
Theoremsetindft 13193* Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
 |-  ( A. x F/ y ph  ->  ( A. x (
 A. y  e.  x  [ y  /  x ] ph  ->  ph )  ->  A. x ph ) )
 
Theoremsetindf 13194* Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
 |-  F/ y ph   =>    |-  ( A. x (
 A. y  e.  x  [ y  /  x ] ph  ->  ph )  ->  A. x ph )
 
Theoremsetindis 13195* Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
 |-  F/ x ps   &    |-  F/ x ch   &    |-  F/ y ph   &    |-  F/ y ps   &    |-  ( x  =  z  ->  ( ph  ->  ps )
 )   &    |-  ( x  =  y 
 ->  ( ch  ->  ph )
 )   =>    |-  ( A. y (
 A. z  e.  y  ps  ->  ch )  ->  A. x ph )
 
Axiomax-bdsetind 13196* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
 |- BOUNDED  ph   =>    |-  ( A. a (
 A. y  e.  a  [ y  /  a ] ph  ->  ph )  ->  A. a ph )
 
Theorembdsetindis 13197* Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  ph   &    |-  F/ x ps   &    |-  F/ x ch   &    |-  F/ y ph   &    |-  F/ y ps   &    |-  ( x  =  z  ->  ( ph  ->  ps ) )   &    |-  ( x  =  y  ->  ( ch  ->  ph ) )   =>    |-  ( A. y ( A. z  e.  y  ps  ->  ch )  ->  A. x ph )
 
Theorembj-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.)
 |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> Ind  A )
 
Theorembj-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.)
 |-  ( A. x  e.  A  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  ->  (Ind  Z  ->  A. u (
 A. t  e.  u  ( t  e.  A  ->  t  e.  Z ) 
 ->  ( u  e.  A  ->  u  e.  Z ) ) ) )
 
Theorembj-inf2vnlem3 13200* Lemma for bj-inf2vn 13202. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |- BOUNDED  Z   =>    |-  ( A. x  e.  A  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  ->  (Ind  Z  ->  A  C_  Z ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13280
  Copyright terms: Public domain < Previous  Next >