HomeHome Intuitionistic Logic Explorer
Theorem List (p. 136 of 137)
< 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 - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
11.2.9.2  Inductive classes and the class of natural number ordinals
 
Syntaxwind 13501 Syntax for inductive classes.
 wff Ind  A
 
Definitiondf-bj-ind 13502* 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 13503 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 13504 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
 |-  ( A  =  B  ->  (Ind 
 A 
 <-> Ind 
 B ) )
 
Theorembj-bdind 13505 Boundedness of the formula "the setvar  x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
 |- BOUNDED Ind  x
 
Theorembj-indint 13506* 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 13507* 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 13508 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 13509  om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
 |- Ind  om
 
Theorembj-omssind 13510  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 13511* 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 13512* 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 13513* Two formulations of the axiom of infinity (see ax-infvn 13516 and bj-omex 13517) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
 |-  ( om  e.  _V  <->  E. x (Ind  x  /\  A. y (Ind  y  ->  x  C_  y )
 ) )
 
11.2.9.3  The first three Peano postulates

The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4553 and peano3 4555 already show this. In this section, we prove bj-peano2 13514 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

 
Theorembj-peano2 13514 Constructive proof of peano2 4554. Temporary note: another possibility is to simply replace sucexg 4457 with bj-sucexg 13497 in the proof of peano2 4554. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  suc  A  e.  om )
 
Theorempeano5set 13515* Version of peano5 4557 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 ) )
 
11.2.10  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.

 
11.2.10.1  The set of natural number ordinals

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13516) and deduce that the class  om of natural number ordinals is a set (bj-omex 13517).

 
Axiomax-infvn 13516* 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 4547) from which one then proves, using full separation, that the wanted set exists (omex 4552). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
 |-  E. x (Ind  x  /\  A. y
 (Ind  y  ->  x  C_  y ) )
 
Theorembj-omex 13517 Proof of omex 4552 from ax-infvn 13516. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
 |-  om  e.  _V
 
11.2.10.2  Peano's fifth postulate

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

 
Theorembdpeano5 13518* Bounded version of peano5 4557. (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 13519* Version of peano5 4557 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 )
 
11.2.10.3  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 natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.

 
Theoremfindset 13520* Bounded induction (principle of induction when  A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4558 for a nonconstructive proof of the general case. See bdfind 13521 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 13521* Bounded induction (principle of induction when  A is assumed to be bounded), proved from basic constructive axioms. See find 4558 for a nonconstructive proof of the general case. See findset 13520 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 13522* 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 4559 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4559, finds2 4560, finds1 4561. (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 13523* Version of bj-bdfindis 13522 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13522 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 13524 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13522 for explanations. From this version, it is easy to prove the bounded version of findes 4562. (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 13525* Constructive proof of a variant of nn0suc 4563. For a constructive proof of nn0suc 4563, see bj-nn0suc 13539. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
 
Theorembj-nntrans 13526 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 13527 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  Tr  A )
 
Theorembj-nnelirr 13528 A natural number does not belong to itself. Version of elirr 4500 for natural numbers, which does not require ax-setind 4496. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  -.  A  e.  A )
 
Theorembj-nnen2lp 13529 A version of en2lp 4513 for natural numbers, which does not require ax-setind 4496.

Note: using this theorem and bj-nnelirr 13528, one can remove dependency on ax-setind 4496 from nntri2 6441 and nndcel 6447; 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 13530 Remove from peano4 4556 dependency on ax-setind 4496. 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 13531 The set  om is transitive. A natural number is included in  om. Constructive proof of elnn 4565.

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 13532 The set  om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  Tr  om
 
Theorembj-nnord 13533 A natural number is an ordinal class. Constructive proof of nnord 4571. Can also be proved from bj-nnelon 13534 if the latter is proved from bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theorembj-nnelon 13534 A natural number is an ordinal. Constructive proof of nnon 4569. Can also be proved from bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theorembj-omord 13535 The set  om is an ordinal class. Constructive proof of ordom 4566. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  Ord  om
 
Theorembj-omelon 13536 The set  om is an ordinal. Constructive proof of omelon 4568. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
 |-  om  e.  On
 
Theorembj-omsson 13537 Constructive proof of omsson 4572. See also bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
 |-  om  C_ 
 On
 
Theorembj-omssonALT 13538 Alternate proof of bj-omsson 13537. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  om  C_ 
 On
 
Theorembj-nn0suc 13539* Proof of (biconditional form of) nn0suc 4563 from the core axioms of CZF. See also bj-nn0sucALT 13553. 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 ) )
 
11.2.11  CZF: Set induction

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

 
11.2.11.1  Set induction

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

 
Theoremsetindft 13540* Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness 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 13541* Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
 |-  F/ y ph   =>    |-  ( A. x (
 A. y  e.  x  [ y  /  x ] ph  ->  ph )  ->  A. x ph )
 
Theoremsetindis 13542* 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 13543* 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 13544* 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 13545* Lemma for bj-inf2vn 13549. 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 13546* Lemma for bj-inf2vnlem3 13547 and bj-inf2vnlem4 13548. 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 13547* Lemma for bj-inf2vn 13549. (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 13548* Lemma for bj-inf2vn2 13550. (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 13549* A sufficient condition for  om to be a set. See bj-inf2vn2 13550 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 13550* A sufficient condition for  om to be a set; unbounded version of bj-inf2vn 13549. (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 13551* Another axiom of infinity in a constructive setting (see ax-infvn 13516). (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 13552 Using bounded set induction and the strong axiom of infinity,  om is a set, that is, we recover ax-infvn 13516 (see bj-2inf 13513 for the equivalence of the latter with bj-omex 13517). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  om  e.  _V
 
Theorembj-nn0sucALT 13553* Alternate proof of bj-nn0suc 13539, also constructive but from ax-inf2 13551, hence requiring ax-bdsetind 13543. (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 ) )
 
11.2.11.2  Full induction

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

 
Theorembj-findis 13554* 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 13522 for a bounded version not requiring ax-setind 4496. See finds 4559 for a proof in IZF. From this version, it is easy to prove of finds 4559, finds2 4560, finds1 4561. (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 13555* Version of bj-findis 13554 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13554 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 13556 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13554 for explanations. From this version, it is easy to prove findes 4562. (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 )
 
11.2.12  CZF: Strong collection

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

 
Axiomax-strcoll 13557* Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that  ph represents a multivalued function on  a, or equivalently a collection of nonempty classes indexed by  a, and the axiom asserts the existence of a set  b which "collects" at least one element in the image of each  x  e.  a and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4079. (Contributed by BJ, 5-Oct-2019.)
 |-  A. a
 ( A. x  e.  a  E. y ph  ->  E. b
 ( A. x  e.  a  E. y  e.  b  ph 
 /\  A. y  e.  b  E. x  e.  a  ph ) )
 
Theoremstrcoll2 13558* Version of ax-strcoll 13557 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. x  e.  a  E. y  e.  b  ph 
 /\  A. y  e.  b  E. x  e.  a  ph ) )
 
Theoremstrcollnft 13559* Closed form of strcollnf 13560. (Contributed by BJ, 21-Oct-2019.)
 |-  ( A. x A. y F/ b ph  ->  ( A. x  e.  a  E. y ph  ->  E. b
 ( A. x  e.  a  E. y  e.  b  ph 
 /\  A. y  e.  b  E. x  e.  a  ph ) ) )
 
Theoremstrcollnf 13560* Version of ax-strcoll 13557 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 13558 with the disjoint variable condition on  b , 
ph replaced with a nonfreeness hypothesis.

This proof aims to demonstrate a standard technique, but strcoll2 13558 will generally suffice: since the theorem asserts the existence of a set  b, supposing that that setvar does not occur in the already defined  ph is not a big constraint. (Contributed by BJ, 21-Oct-2019.)

 |-  F/ b ph   =>    |-  ( A. x  e.  a  E. y ph  ->  E. b ( A. x  e.  a  E. y  e.  b  ph  /\ 
 A. y  e.  b  E. x  e.  a  ph ) )
 
TheoremstrcollnfALT 13561* Alternate proof of strcollnf 13560, not using strcollnft 13559. (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. x  e.  a  E. y  e.  b  ph  /\ 
 A. y  e.  b  E. x  e.  a  ph ) )
 
11.2.13  CZF: Subset collection

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

 
Axiomax-sscoll 13562* Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that  ph represents a multivalued function from  a to  b, or equivalently a collection of nonempty subsets of  b indexed by  a, and the consequent asserts the existence of a subset of  c which "collects" at least one element in the image of each  x  e.  a and which is made only of such elements. The axiom asserts the existence, for any sets  a ,  b, of a set  c such that that implication holds for any value of the parameter  z of  ph. (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. x  e.  a  E. y  e.  d  ph  /\ 
 A. y  e.  d  E. x  e.  a  ph ) )
 
Theoremsscoll2 13563* Version of ax-sscoll 13562 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. x  e.  a  E. y  e.  d  ph  /\ 
 A. y  e.  d  E. x  e.  a  ph ) )
 
11.2.14  Real numbers
 
Axiomax-ddkcomp 13564 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 Axiom ax-ddkcomp 13564 should be used in place of construction specific results. In particular, axcaucvg 7820 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
 
Theoremss1oel2o 13565 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4159 which more directly illustrates the contrast with el2oss1o 6390. (Contributed by Jim Kingdon, 8-Aug-2022.)
 |-  (EXMID  <->  A. x ( x 
 C_  1o  ->  x  e. 
 2o ) )
 
Theoremnnti 13566 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 ) ) )
 
Theorem012of 13567 Mapping zero and one between  NN0 and  om style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
 |-  G  = frec ( ( x  e. 
 ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( `' G  |`  { 0 ,  1 } ) : { 0 ,  1 } --> 2o
 
Theorem2o01f 13568 Mapping zero and one between  om and  NN0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
 |-  G  = frec ( ( x  e. 
 ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( G  |`  2o ) : 2o --> { 0 ,  1 }
 
11.3.2  The power set of a singleton
 
Theorempwtrufal 13569 A subset of the singleton  { (/) } cannot be anything other than  (/) or  { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4159. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4157), 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 13570* 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 13571* 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 13572* 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 13573* 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 )
 
Theoremsssneq 13574* Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
 |-  ( A  C_  { B }  ->  A. y  e.  A  A. z  e.  A  y  =  z )
 
Theorempw1nct 13575* A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
 |-  ( A. r ( r  C_  ( ~P 1o  X.  om )  ->  ( A. p  e.  ~P  1o E. n  e.  om  p r n 
 ->  E. m  e.  om  A. q  e.  ~P  1o q r m ) )  ->  -.  E. f  f : om -onto-> ( ~P 1o 1o ) )
 
11.3.3  Omniscience of NN+oo
 
Theorem0nninf 13576 The zero element of ℕ (the constant sequence equal to  (/)). (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  ( om  X.  { (/) } )  e.
 
Theoremnnsf 13577* Domain and range of  S. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
 |-  S  =  ( p  e. 
 |->  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( p `
  U. i ) ) ) )   =>    |-  S : -->
 
Theorempeano4nninf 13578* The successor function on ℕ is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
 |-  S  =  ( p  e. 
 |->  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( p `
  U. i ) ) ) )   =>    |-  S : -1-1->
 
Theorempeano3nninf 13579* The successor function on ℕ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
 |-  S  =  ( p  e. 
 |->  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( p `
  U. i ) ) ) )   =>    |-  ( A  e.  ->  ( S `  A )  =/=  ( x  e.  om  |->  (/) ) )
 
Theoremnninfalllem1 13580* Lemma for nninfall 13581. (Contributed by Jim Kingdon, 1-Aug-2022.)
 |-  ( ph  ->  Q  e.  ( 2o  ^m ) )   &    |-  ( ph  ->  ( Q `  ( x  e.  om  |->  1o )
 )  =  1o )   &    |-  ( ph  ->  A. n  e.  om  ( Q `  ( i  e.  om  |->  if (
 i  e.  n ,  1o ,  (/) ) ) )  =  1o )   &    |-  ( ph  ->  P  e. )   &    |-  ( ph  ->  ( Q `  P )  =  (/) )   =>    |-  ( ph  ->  A. n  e.  om  ( P `  n )  =  1o )
 
Theoremnninfall 13581* Given a decidable predicate on ℕ, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which  Q is a decidable predicate is that it assigns a value of either  (/) or  1o (which can be thought of as false and true) to every element of ℕ. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
 |-  ( ph  ->  Q  e.  ( 2o  ^m ) )   &    |-  ( ph  ->  ( Q `  ( x  e.  om  |->  1o )
 )  =  1o )   &    |-  ( ph  ->  A. n  e.  om  ( Q `  ( i  e.  om  |->  if (
 i  e.  n ,  1o ,  (/) ) ) )  =  1o )   =>    |-  ( ph  ->  A. p  e.  ( Q `  p )  =  1o )
 
Theoremnninfsellemdc 13582* Lemma for nninfself 13585. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
 |-  (
 ( Q  e.  ( 2o  ^m )  /\  N  e.  om )  -> DECID  A. k  e.  suc  N ( Q `  (
 i  e.  om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o )
 
Theoremnninfsellemcl 13583* Lemma for nninfself 13585. (Contributed by Jim Kingdon, 8-Aug-2022.)
 |-  (
 ( Q  e.  ( 2o  ^m )  /\  N  e.  om )  ->  if ( A. k  e.  suc  N ( Q `  ( i  e.  om  |->  if (
 i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) )  e.  2o )
 
Theoremnninfsellemsuc 13584* Lemma for nninfself 13585. (Contributed by Jim Kingdon, 6-Aug-2022.)
 |-  (
 ( Q  e.  ( 2o  ^m )  /\  J  e.  om )  ->  if ( A. k  e.  suc  suc  J ( Q `  ( i  e.  om  |->  if (
 i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) )  C_  if ( A. k  e.  suc  J ( Q `  (
 i  e.  om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) )
 
Theoremnninfself 13585* Domain and range of the selection function for ℕ. (Contributed by Jim Kingdon, 6-Aug-2022.)
 |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
 om  |->  if ( A. k  e.  suc  n ( q `
  ( i  e. 
 om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )   =>    |-  E : ( 2o  ^m ) -->
 
Theoremnninfsellemeq 13586* Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.)
 |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
 om  |->  if ( A. k  e.  suc  n ( q `
  ( i  e. 
 om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )   &    |-  ( ph  ->  Q  e.  ( 2o  ^m ) )   &    |-  ( ph  ->  ( Q `  ( E `
  Q ) )  =  1o )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  A. k  e.  N  ( Q `  ( i  e.  om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o )   &    |-  ( ph  ->  ( Q `  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )  =  (/) )   =>    |-  ( ph  ->  ( E `  Q )  =  ( i  e. 
 om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )
 
Theoremnninfsellemqall 13587* Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.)
 |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
 om  |->  if ( A. k  e.  suc  n ( q `
  ( i  e. 
 om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )   &    |-  ( ph  ->  Q  e.  ( 2o  ^m ) )   &    |-  ( ph  ->  ( Q `  ( E `
  Q ) )  =  1o )   &    |-  ( ph  ->  N  e.  om )   =>    |-  ( ph  ->  ( Q `  ( i  e. 
 om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )  =  1o )
 
Theoremnninfsellemeqinf 13588* Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.)
 |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
 om  |->  if ( A. k  e.  suc  n ( q `
  ( i  e. 
 om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )   &    |-  ( ph  ->  Q  e.  ( 2o  ^m ) )   &    |-  ( ph  ->  ( Q `  ( E `
  Q ) )  =  1o )   =>    |-  ( ph  ->  ( E `  Q )  =  ( i  e. 
 om  |->  1o ) )
 
Theoremnninfsel 13589*  E is a selection function for ℕ. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.)
 |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
 om  |->  if ( A. k  e.  suc  n ( q `
  ( i  e. 
 om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )   &    |-  ( ph  ->  Q  e.  ( 2o  ^m ) )   &    |-  ( ph  ->  ( Q `  ( E `
  Q ) )  =  1o )   =>    |-  ( ph  ->  A. p  e.  ( Q `  p )  =  1o )
 
Theoremnninfomnilem 13590* Lemma for nninfomni 13591. (Contributed by Jim Kingdon, 10-Aug-2022.)
 |-  E  =  ( q  e.  ( 2o  ^m )  |->  ( n  e. 
 om  |->  if ( A. k  e.  suc  n ( q `
  ( i  e. 
 om  |->  if ( i  e.  k ,  1o ,  (/) ) ) )  =  1o ,  1o ,  (/) ) ) )   =>    |-  e. Omni
 
Theoremnninfomni 13591 is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
 |-  e. Omni
 
Theoremnninffeq 13592* Equality of two functions on ℕ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one,  |-  ( ph  ->  A. n  e.  suc  om
... ). (Contributed by Jim Kingdon, 4-Aug-2023.)
 |-  ( ph  ->  F : --> NN0 )   &    |-  ( ph  ->  G : --> NN0 )   &    |-  ( ph  ->  ( F `  ( x  e.  om  |->  1o )
 )  =  ( G `
  ( x  e. 
 om  |->  1o ) ) )   &    |-  ( ph  ->  A. n  e. 
 om  ( F `  ( i  e.  om  |->  if ( i  e.  n ,  1o ,  (/) ) ) )  =  ( G `
  ( i  e. 
 om  |->  if ( i  e.  n ,  1o ,  (/) ) ) ) )   =>    |-  ( ph  ->  F  =  G )
 
11.3.4  Schroeder-Bernstein Theorem
 
Theoremexmidsbthrlem 13593* Lemma for exmidsbthr 13594. (Contributed by Jim Kingdon, 11-Aug-2022.)
 |-  S  =  ( p  e. 
 |->  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( p `
  U. i ) ) ) )   =>    |-  ( A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y )  -> EXMID )
 
Theoremexmidsbthr 13594* The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
 |-  ( A. x A. y ( ( x  ~<_  y  /\  y 
 ~<_  x )  ->  x  ~~  y )  -> EXMID )
 
Theoremexmidsbth 13595* The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6911) is the proof of the Schroeder-Bernstein Theorem from the Metamath Proof Explorer database (in which excluded middle holds), but adapted to use EXMID as an antecedent rather than being unconditionally true, as in the non-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 6911.

The reverse direction (exmidsbthr 13594) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)

 |-  (EXMID  <->  A. x A. y
 ( ( x  ~<_  y 
 /\  y  ~<_  x ) 
 ->  x  ~~  y ) )
 
Theoremsbthomlem 13596 Lemma for sbthom 13597. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
 |-  ( ph  ->  om  e. Omni )   &    |-  ( ph  ->  Y  C_  { (/) } )   &    |-  ( ph  ->  F : om -1-1-onto-> ( Y om ) )   =>    |-  ( ph  ->  ( Y  =  (/)  \/  Y  =  { (/) } ) )
 
Theoremsbthom 13597 Schroeder-Bernstein is not possible even for  om. We know by exmidsbth 13595 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is  om? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
 |-  (
 ( A. x ( ( x  ~<_  om  /\  om  ~<_  x ) 
 ->  x  ~~  om )  /\  om  e. Omni )  -> EXMID )
 
11.3.5  Real and complex numbers
 
Theoremqdencn 13598* The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 11102 (and also would hold for  RR  X.  RR with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  Q  =  { z  e.  CC  |  ( ( Re `  z )  e.  QQ  /\  ( Im `  z
 )  e.  QQ ) }   =>    |-  ( ( A  e.  CC  /\  B  e.  RR+ )  ->  E. x  e.  Q  ( abs `  ( x  -  A ) )  <  B )
 
Theoremrefeq 13599* Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  G : RR --> RR )   &    |-  ( ph  ->  A. x  e.  RR  ( x  <  0  ->  ( F `  x )  =  ( G `  x ) ) )   &    |-  ( ph  ->  A. x  e. 
 RR  ( 0  < 
 x  ->  ( F `  x )  =  ( G `  x ) ) )   &    |-  ( ph  ->  ( F `  0 )  =  ( G `  0 ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremtriap 13600 Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <  B  \/  A  =  B  \/  B  <  A )  <-> DECID  A #  B ) )
    < 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 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13652
  Copyright terms: Public domain < Previous  Next >