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Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdinex1g 12901 Bounded version of inex1g 4032. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   =>    |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theorembdssex 12902 Bounded version of ssex 4033. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theorembdssexi 12903 Bounded version of ssexi 4034. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |-  B  e.  _V   &    |-  A  C_  B   =>    |-  A  e.  _V
 
Theorembdssexg 12904 Bounded version of ssexg 4035. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   =>    |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
 
Theorembdssexd 12905 Bounded version of ssexd 4036. (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 12906* Bounded version of rabexg 4039. (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 12907 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 12908 intexr 4043 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
 
Theorembj-intnexr 12909 intnexr 4044 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( |^| A  =  _V  ->  -. 
 |^| A  e.  _V )
 
Theorembj-zfpair2 12910 Proof of zfpair2 4100 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  { x ,  y }  e.  _V
 
Theorembj-prexg 12911 Proof of prexg 4101 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 12912 snexg 4076 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { A }  e.  _V )
 
Theorembj-snex 12913 snex 4077 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 { A }  e.  _V
 
Theorembj-sels 12914* 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 12915* axun2 4325 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 12916* uniex2 4326 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y  y  =  U. x
 
Theorembj-uniex 12917 uniex 4327 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 U. A  e.  _V
 
Theorembj-uniexg 12918 uniexg 4329 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  U. A  e.  _V )
 
Theorembj-unex 12919 unex 4330 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 12920 Bounded version of unexb 4331. (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 12921 unexg 4332 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 12922 sucexg 4382 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  suc 
 A  e.  _V )
 
Theorembj-sucex 12923 sucex 4383 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 12924 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
 |- BOUNDED  ph   =>    |- DECID  ph
 
Theorembj-d0clsepcl 12925 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 12926 Syntax for inductive classes.
 wff Ind  A
 
Definitiondf-bj-ind 12927* 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 12928 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 12929 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
 |-  ( A  =  B  ->  (Ind 
 A 
 <-> Ind 
 B ) )
 
Theorembj-bdind 12930 Boundedness of the formula "the setvar  x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
 |- BOUNDED Ind  x
 
Theorembj-indint 12931* 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 12932* 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 12933 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 12934  om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
 |- Ind  om
 
Theorembj-omssind 12935  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 12936* 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 12937* 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 12938* Two formulations of the axiom of infinity (see ax-infvn 12941 and bj-omex 12942) . (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 4476 and peano3 4478 already show this. In this section, we prove bj-peano2 12939 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

 
Theorembj-peano2 12939 Constructive proof of peano2 4477. Temporary note: another possibility is to simply replace sucexg 4382 with bj-sucexg 12922 in the proof of peano2 4477. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  suc  A  e.  om )
 
Theorempeano5set 12940* Version of peano5 4480 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 12941) and deduce that the class  om of finite ordinals is a set (bj-omex 12942).

 
Axiomax-infvn 12941* 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 4470) from which one then proves, using full separation, that the wanted set exists (omex 4475). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
 |-  E. x (Ind  x  /\  A. y
 (Ind  y  ->  x  C_  y ) )
 
Theorembj-omex 12942 Proof of omex 4475 from ax-infvn 12941. (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 12943* Bounded version of peano5 4480. (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 12944* Version of peano5 4480 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 12945* Bounded induction (principle of induction when  A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4481 for a nonconstructive proof of the general case. See bdfind 12946 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 12946* Bounded induction (principle of induction when  A is assumed to be bounded), proved from basic constructive axioms. See find 4481 for a nonconstructive proof of the general case. See findset 12945 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 12947* 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 4482 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4482, finds2 4483, finds1 4484. (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 12948* Version of bj-bdfindis 12947 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 12947 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 12949 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 12947 for explanations. From this version, it is easy to prove the bounded version of findes 4485. (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 12950* Constructive proof of a variant of nn0suc 4486. For a constructive proof of nn0suc 4486, see bj-nn0suc 12964. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
 
Theorembj-nntrans 12951 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 12952 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  Tr  A )
 
Theorembj-nnelirr 12953 A natural number does not belong to itself. Version of elirr 4424 for natural numbers, which does not require ax-setind 4420. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  -.  A  e.  A )
 
Theorembj-nnen2lp 12954 A version of en2lp 4437 for natural numbers, which does not require ax-setind 4420.

Note: using this theorem and bj-nnelirr 12953, one can remove dependency on ax-setind 4420 from nntri2 6356 and nndcel 6362; 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 12955 Remove from peano4 4479 dependency on ax-setind 4420. 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 12956 The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4487.

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 12957 The set  om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  Tr  om
 
Theorembj-nnord 12958 A natural number is an ordinal. Constructive proof of nnord 4493. Can also be proved from bj-nnelon 12959 if the latter is proved from bj-omssonALT 12963. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theorembj-nnelon 12959 A natural number is an ordinal. Constructive proof of nnon 4491. Can also be proved from bj-omssonALT 12963. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theorembj-omord 12960 The set  om is an ordinal. Constructive proof of ordom 4488. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  Ord  om
 
Theorembj-omelon 12961 The set  om is an ordinal. Constructive proof of omelon 4490. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  om  e.  On
 
Theorembj-omsson 12962 Constructive proof of omsson 4494. See also bj-omssonALT 12963. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
 |-  om  C_ 
 On
 
Theorembj-omssonALT 12963 Alternate proof of bj-omsson 12962. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  om  C_ 
 On
 
Theorembj-nn0suc 12964* Proof of (biconditional form of) nn0suc 4486 from the core axioms of CZF. See also bj-nn0sucALT 12978. 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 12965* 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 12966* 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 12967* 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 12968* 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 12969* 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 12970* Lemma for bj-inf2vn 12974. 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 12971* Lemma for bj-inf2vnlem3 12972 and bj-inf2vnlem4 12973. 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 12972* Lemma for bj-inf2vn 12974. (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 ) )
 
Theorembj-inf2vnlem4 12973* Lemma for bj-inf2vn2 12975. (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  C_  Z ) )
 
Theorembj-inf2vn 12974* A sufficient condition for  om to be a set. See bj-inf2vn2 12975 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   =>    |-  ( A  e.  V  ->  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A  =  om )
 )
 
Theorembj-inf2vn2 12975* A sufficient condition for  om to be a set; unbounded version of bj-inf2vn 12974. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  (
 A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A  =  om )
 )
 
Axiomax-inf2 12976* Another axiom of infinity in a constructive setting (see ax-infvn 12941). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
 |-  E. a A. x ( x  e.  a  <->  ( x  =  (/)  \/  E. y  e.  a  x  =  suc  y ) )
 
Theorembj-omex2 12977 Using bounded set induction and the strong axiom of infinity,  om is a set, that is, we recover ax-infvn 12941 (see bj-2inf 12938 for the equivalence of the latter with bj-omex 12942). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  om  e.  _V
 
Theorembj-nn0sucALT 12978* Alternate proof of bj-nn0suc 12964, also constructive but from ax-inf2 12976, hence requiring ax-bdsetind 12968. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  om  <->  ( A  =  (/) 
 \/  E. x  e.  om  A  =  suc  x ) )
 
11.2.10.2  Full induction

In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.

 
Theorembj-findis 12979* Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 12947 for a bounded version not requiring ax-setind 4420. See finds 4482 for a proof in IZF. From this version, it is easy to prove of finds 4482, finds2 4483, finds1 4484. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
 |-  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-findisg 12980* Version of bj-findis 12979 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 12979 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
 |-  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-findes 12981 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 12979 for explanations. From this version, it is easy to prove findes 4485. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
 |-  (
 ( [. (/)  /  x ]. ph 
 /\  A. x  e.  om  ( ph  ->  [. suc  x  /  x ]. ph )
 )  ->  A. x  e. 
 om  ph )
 
11.2.11  CZF: Strong collection

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

 
Axiomax-strcoll 12982* Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.)
 |-  A. a
 ( A. x  e.  a  E. y ph  ->  E. b A. y ( y  e.  b  <->  E. x  e.  a  ph ) )
 
Theoremstrcoll2 12983* Version of ax-strcoll 12982 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
 |-  ( A. x  e.  a  E. y ph  ->  E. b A. y ( y  e.  b  <->  E. x  e.  a  ph ) )
 
Theoremstrcollnft 12984* Closed form of strcollnf 12985. Version of ax-strcoll 12982 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
 |-  ( A. x A. y F/ b ph  ->  ( A. x  e.  a  E. y ph  ->  E. b A. y ( y  e.  b  <->  E. x  e.  a  ph ) ) )
 
Theoremstrcollnf 12985* Version of ax-strcoll 12982 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
 |-  F/ b ph   =>    |-  ( A. x  e.  a  E. y ph  ->  E. b A. y
 ( y  e.  b  <->  E. x  e.  a  ph ) )
 
TheoremstrcollnfALT 12986* Alternate proof of strcollnf 12985, not using strcollnft 12984. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F/ b ph   =>    |-  ( A. x  e.  a  E. y ph  ->  E. b A. y
 ( y  e.  b  <->  E. x  e.  a  ph ) )
 
11.2.12  CZF: Subset collection

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

 
Axiomax-sscoll 12987* Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.)
 |-  A. a A. b E. c A. z ( A. x  e.  a  E. y  e.  b  ph  ->  E. d  e.  c  A. y ( y  e.  d  <->  E. x  e.  a  ph ) )
 
Theoremsscoll2 12988* Version of ax-sscoll 12987 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
 |-  E. c A. z ( A. x  e.  a  E. y  e.  b  ph  ->  E. d  e.  c  A. y ( y  e.  d  <->  E. x  e.  a  ph ) )
 
11.2.13  Real numbers
 
Axiomax-ddkcomp 12989 Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 12989 should be used in place of construction specific results. In particular, axcaucvg 7672 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
 |-  (
 ( ( A  C_  RR  /\  E. x  x  e.  A )  /\  E. x  e.  RR  A. y  e.  A  y  <  x  /\  A. x  e.  RR  A. y  e. 
 RR  ( x  < 
 y  ->  ( E. z  e.  A  x  <  z  \/  A. z  e.  A  z  <  y
 ) ) )  ->  E. x  e.  RR  ( A. y  e.  A  y  <_  x  /\  (
 ( B  e.  R  /\  A. y  e.  A  y  <_  B )  ->  x  <_  B ) ) )
 
11.3  Mathbox for Jim Kingdon
 
11.3.1  Natural numbers
 
Theoremel2oss1o 12990 Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 12991. (Contributed by Jim Kingdon, 8-Aug-2022.)
 |-  ( A  e.  2o  ->  A 
 C_  1o )
 
Theoremss1oel2o 12991 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4089 which more directly illustrates the contrast with el2oss1o 12990. (Contributed by Jim Kingdon, 8-Aug-2022.)
 |-  (EXMID  <->  A. x ( x 
 C_  1o  ->  x  e. 
 2o ) )
 
Theorempw1dom2 12992 The power set of  1o dominates  2o. Also see pwpw0ss 3699 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
 |-  2o  ~<_  ~P 1o
 
Theoremnnti 12993 Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.)
 |-  ( ph  ->  A  e.  om )   =>    |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u  _E  v  /\  -.  v  _E  u ) ) )
 
11.3.2  The power set of a singleton
 
Theorempwtrufal 12994 A subset of the singleton  { (/) } cannot be anything other than  (/) or  { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4089. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4088), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
 |-  ( A  C_  { (/) }  ->  -. 
 -.  ( A  =  (/) 
 \/  A  =  { (/)
 } ) )
 
Theorempwle2 12995* An exercise related to  N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
 |-  T  =  U_ x  e.  N  ( { x }  X.  1o )   =>    |-  ( ( N  e.  om 
 /\  G : T -1-1-> ~P 1o )  ->  N  C_ 
 2o )
 
Theorempwf1oexmid 12996* An exercise related to  N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
 |-  T  =  U_ x  e.  N  ( { x }  X.  1o )   =>    |-  ( ( N  e.  om 
 /\  G : T -1-1-> ~P 1o )  ->  ( ran  G  =  ~P 1o  <->  ( N  =  2o  /\ EXMID ) ) )
 
Theoremexmid1stab 12997* If any proposition is stable, excluded middle follows. We are thinking of  x as a proposition and  x  =  { (/)
} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
 |-  (
 ( ph  /\  x  C_  { (/) } )  -> STAB  x  =  { (/)
 } )   =>    |-  ( ph  -> EXMID )
 
Theoremsubctctexmid 12998* If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
 |-  ( ph  ->  A. x ( E. s ( s  C_  om 
 /\  E. f  f : s -onto-> x )  ->  E. g  g : om -onto-> ( x 1o ) ) )   &    |-  ( ph  ->  om  e. Markov )   =>    |-  ( ph  -> EXMID )
 
11.3.3  Omniscience of NN+oo
 
Theorem0nninf 12999 The zero element of ℕ (the constant sequence equal to  (/)). (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  ( om  X.  { (/) } )  e.
 
Theoremnninff 13000 An element of ℕ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( A  e.  ->  A : om --> 2o )
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