TOC of Theorem List - Intuitionistic Logic Explorer

Table of Contents Summary

PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Predicate calculus mostly without distinct variables
      1.4  Predicate calculus with distinct variables
      1.5  First-order logic with one non-logical binary predicate
PART 2  SET THEORY
      2.1  IZF Set Theory - start with the Axiom of Extensionality
      2.2  IZF Set Theory - add the Axioms of Collection and Separation
      2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
      2.4  IZF Set Theory - add the Axiom of Union
      2.5  IZF Set Theory - add the Axiom of Set Induction
      2.6  IZF Set Theory - add the Axiom of Infinity
PART 3  CHOICE PRINCIPLES
      3.1  Countable Choice and Dependent Choice
PART 4  REAL AND COMPLEX NUMBERS
      4.1  Construction and axiomatization of real and complex numbers
      4.2  Derive the basic properties from the field axioms
      4.3  Real and complex numbers - basic operations
      4.4  Integer sets
      4.5  Order sets
      4.6  Elementary integer functions
      4.7  Elementary real and complex functions
      4.8  Elementary limits and convergence
      4.9  Elementary trigonometry
PART 5  ELEMENTARY NUMBER THEORY
      5.1  Elementary properties of divisibility
      5.2  Elementary prime number theory
      5.3  Cardinality of real and complex number subsets
PART 6  BASIC STRUCTURES
      6.1  Extensible structures
PART 7  BASIC ALGEBRAIC STRUCTURES
      7.1  Monoids
      7.2  Groups
      7.3  Rings
      7.4  The complex numbers as an algebraic extensible structure
PART 8  BASIC TOPOLOGY
      8.1  Topology
      8.2  Metric spaces
PART 9  BASIC REAL AND COMPLEX ANALYSIS
      9.1  Derivatives
PART 10  BASIC REAL AND COMPLEX FUNCTIONS
      10.1  Basic trigonometry
      10.2  Basic number theory
PART 11  GUIDES AND MISCELLANEA
      11.1  Guides (conventions, explanations, and examples)
PART 12  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​12.1  Mathboxes for user contributions
      12.2  Mathbox for BJ
      12.3  Mathbox for Jim Kingdon
      12.4  Mathbox for Mykola Mostovenko
      12.5  Mathbox for David A. Wheeler

Detailed Table of Contents
(* means the section header has a description)

​​*PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            1.2.2  Propositional logic axioms for implication   ax-mp 5
            ​*1.2.3  Logical implication   mp2b 8
            1.2.4  Logical conjunction and logical equivalence   wa 104
            1.2.5  Logical negation (intuitionistic)   ax-in1 614
            1.2.6  Logical disjunction   wo 708
            1.2.7  Stable propositions   wstab 830
            1.2.8  Decidable propositions   wdc 834
            ​*1.2.9  Theorems of decidable propositions   const 852
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 916
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 977
            1.2.12  True and false constants   wal 1351
                  ​*1.2.12.1  Universal quantifier for use by df-tru   wal 1351
                  ​*1.2.12.2  Equality predicate for use by df-tru   cv 1352
                  1.2.12.3  Define the true and false constants   wtru 1354
            1.2.13  Logical 'xor'   wxo 1375
            ​*1.2.14  Truth tables: Operations on true and false constants   truantru 1401
            ​*1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1423
            1.2.16  Logical implication (continued)   syl6an 1434
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1447
            ​*1.3.2  Equality predicate (continued)   weq 1503
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1526
            1.3.4  Introduce Axiom of Existence   ax-i9 1530
            1.3.5  Additional intuitionistic axioms   ax-ial 1534
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1536
            1.3.7  The existential quantifier   19.8a 1590
            1.3.8  Equality theorems without distinct variables   a9e 1696
            1.3.9  Axioms ax-10 and ax-11   ax10o 1715
            1.3.10  Substitution (without distinct variables)   wsb 1762
            1.3.11  Theorems using axiom ax-11   equs5a 1794
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1811
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1822
            1.4.3  More theorems related to ax-11 and substitution   albidv 1824
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1858
            1.4.5  More substitution theorems   hbs1 1938
            1.4.6  Existential uniqueness   weu 2026
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2124
      ​​*1.5  First-order logic with one non-logical binary predicate
​​*PART 2  SET THEORY
      ​2.1  IZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2159
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2163
            2.1.3  Class form not-free predicate   wnfc 2306
            2.1.4  Negated equality and membership   wne 2347
                  2.1.4.1  Negated equality   wne 2347
                  2.1.4.2  Negated membership   wnel 2442
            2.1.5  Restricted quantification   wral 2455
            2.1.6  The universal class   cvv 2737
            ​*2.1.7  Conditional equality (experimental)   wcdeq 2945
            2.1.8  Russell's Paradox   ru 2961
            2.1.9  Proper substitution of classes for sets   wsbc 2962
            2.1.10  Proper substitution of classes for sets into classes   csb 3057
            2.1.11  Define basic set operations and relations   cdif 3126
            2.1.12  Subclasses and subsets   df-ss 3142
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3245
                  2.1.13.1  The difference of two classes   dfdif3 3245
                  2.1.13.2  The union of two classes   elun 3276
                  2.1.13.3  The intersection of two classes   elin 3318
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3366
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3401
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3415
            2.1.14  The empty set   c0 3422
            2.1.15  Conditional operator   cif 3534
            2.1.16  Power classes   cpw 3574
            2.1.17  Unordered and ordered pairs   csn 3591
            2.1.18  The union of a class   cuni 3807
            2.1.19  The intersection of a class   cint 3842
            2.1.20  Indexed union and intersection   ciun 3884
            2.1.21  Disjointness   wdisj 3977
            2.1.22  Binary relations   wbr 4000
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4060
            2.1.24  Transitive classes   wtr 4098
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4115
            2.2.2  Introduce the Axiom of Separation   ax-sep 4118
            2.2.3  Derive the Null Set Axiom   zfnuleu 4124
            2.2.4  Theorems requiring subset and intersection existence   nalset 4130
            2.2.5  Theorems requiring empty set existence   class2seteq 4160
            2.2.6  Collection principle   bnd 4169
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4171
            2.3.2  A notation for excluded middle   wem 4191
            2.3.3  Axiom of Pairing   ax-pr 4206
            2.3.4  Ordered pair theorem   opm 4231
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4254
            2.3.6  Power class of union and intersection   pwin 4279
            2.3.7  Epsilon and identity relations   cep 4284
            ​*2.3.8  Partial and total orderings   wpo 4291
            2.3.9  Founded and set-like relations   wfrfor 4324
            2.3.10  Ordinals   word 4359
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4430
            2.4.2  Ordinals (continued)   ordon 4482
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4528
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4533
            2.5.3  Transfinite induction   tfi 4578
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4584
            2.6.2  The natural numbers   com 4586
            2.6.3  Peano's postulates   peano1 4590
            2.6.4  Finite induction (for finite ordinals)   find 4595
            2.6.5  The Natural Numbers (continued)   nn0suc 4600
            2.6.6  Relations   cxp 4621
            2.6.7  Definite description binder (inverted iota)   cio 5172
            2.6.8  Functions   wfun 5206
            2.6.9  Cantor's Theorem   canth 5823
            2.6.10  Restricted iota (description binder)   crio 5824
            2.6.11  Operations   co 5869
            2.6.12  Maps-to notation   elmpocl 6063
            2.6.13  Function operation   cof 6075
            2.6.14  Functions (continued)   resfunexgALT 6103
            2.6.15  First and second members of an ordered pair   c1st 6133
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6233
            2.6.17  Function transposition   ctpos 6239
            2.6.18  Undefined values   pwuninel2 6277
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6279
            2.6.20  "Strong" transfinite recursion   crecs 6299
            2.6.21  Recursive definition generator   crdg 6364
            2.6.22  Finite recursion   cfrec 6385
            2.6.23  Ordinal arithmetic   c1o 6404
            2.6.24  Natural number arithmetic   nna0 6469
            2.6.25  Equivalence relations and classes   wer 6526
            2.6.26  The mapping operation   cmap 6642
            2.6.27  Infinite Cartesian products   cixp 6692
            2.6.28  Equinumerosity   cen 6732
            2.6.29  Equinumerosity (cont.)   xpf1o 6838
            2.6.30  Pigeonhole Principle   phplem1 6846
            2.6.31  Finite sets   fict 6862
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 6950
            2.6.33  Finite intersections   cfi 6961
            2.6.34  Supremum and infimum   csup 6975
            2.6.35  Ordinal isomorphism   ordiso2 7028
            2.6.36  Disjoint union   cdju 7030
                  2.6.36.1  Disjoint union   cdju 7030
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7038
                  2.6.36.3  Universal property of the disjoint union   djuss 7063
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7086
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7095
                  2.6.36.6  Countable sets   0ct 7100
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7112
            2.6.38  Omniscient sets   comni 7126
            2.6.39  Markov's principle   cmarkov 7143
            2.6.40  Weakly omniscient sets   cwomni 7155
            2.6.41  Cardinal numbers   ccrd 7172
            2.6.42  Axiom of Choice equivalents   wac 7198
            2.6.43  Cardinal number arithmetic   endjudisj 7203
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7214
            2.6.45  Excluded middle and the power set of a singleton   pw1on 7219
            2.6.46  Apartness relations   wtap 7240
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7252
​​*PART 4  REAL AND COMPLEX NUMBERS
      ​4.1  Construction and axiomatization of real and complex numbers
            4.1.1  Dedekind-cut construction of real and complex numbers   cnpi 7262
            4.1.2  Final derivation of real and complex number postulates   axcnex 7849
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7893
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 7926
            4.2.2  Infinity and the extended real number system   cpnf 7979
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8014
            4.2.4  Ordering on reals   lttr 8021
            4.2.5  Initial properties of the complex numbers   mul12 8076
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8105
            4.3.2  Subtraction   cmin 8118
            4.3.3  Multiplication   kcnktkm1cn 8330
            4.3.4  Ordering on reals (cont.)   ltadd2 8366
            4.3.5  Real Apartness   creap 8521
            4.3.6  Complex Apartness   cap 8528
            4.3.7  Reciprocals   recextlem1 8597
            4.3.8  Division   cdiv 8618
            4.3.9  Ordering on reals (cont.)   ltp1 8790
            4.3.10  Suprema   lbreu 8891
            4.3.11  Imaginary and complex number properties   crap0 8904
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 8908
            4.4.2  Principle of mathematical induction   nnind 8924
            ​*4.4.3  Decimal representation of numbers   c2 8959
            ​*4.4.4  Some properties of specific numbers   neg1cn 9013
            4.4.5  Simple number properties   halfcl 9134
            4.4.6  The Archimedean property   arch 9162
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9165
            ​*4.4.8  Extended nonnegative integers   cxnn0 9228
            4.4.9  Integers (as a subset of complex numbers)   cz 9242
            4.4.10  Decimal arithmetic   cdc 9373
            4.4.11  Upper sets of integers   cuz 9517
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9608
            4.4.13  Complex numbers as pairs of reals   cnref1o 9639
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9640
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9756
            4.5.3  Real number intervals   cioo 9875
            4.5.4  Finite intervals of integers   cfz 9995
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10098
            4.5.6  Half-open integer ranges   cfzo 10128
            4.5.7  Rational numbers (cont.)   qtri3or 10229
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10254
            4.6.2  The modulo (remainder) operation   cmo 10308
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10385
            4.6.4  Strong induction over upper sets of integers   uzsinds 10428
            4.6.5  The infinite sequence builder "seq"   cseq 10431
            4.6.6  Integer powers   cexp 10505
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10683
            4.6.8  Factorial function   cfa 10689
            4.6.9  The binomial coefficient operation   cbc 10711
            4.6.10  The ` # ` (set size) function   chash 10739
      ​4.7  Elementary real and complex functions
            4.7.1  The "shift" operation   cshi 10807
            4.7.2  Real and imaginary parts; conjugate   ccj 10832
            4.7.3  Sequence convergence   caucvgrelemrec 10972
            4.7.4  Square root; absolute value   csqrt 10989
            4.7.5  The maximum of two real numbers   maxcom 11196
            4.7.6  The minimum of two real numbers   mincom 11221
            4.7.7  The maximum of two extended reals   xrmaxleim 11236
            4.7.8  The minimum of two extended reals   xrnegiso 11254
      ​4.8  Elementary limits and convergence
            4.8.1  Limits   cli 11270
            4.8.2  Finite and infinite sums   csu 11345
            4.8.3  The binomial theorem   binomlem 11475
            4.8.4  Infinite sums (cont.)   isumshft 11482
            4.8.5  Miscellaneous converging and diverging sequences   divcnv 11489
            4.8.6  Arithmetic series   arisum 11490
            4.8.7  Geometric series   expcnvap0 11494
            4.8.8  Ratio test for infinite series convergence   cvgratnnlembern 11515
            4.8.9  Mertens' theorem   mertenslemub 11526
            4.8.10  Finite and infinite products   prodf 11530
                  4.8.10.1  Product sequences   prodf 11530
                  4.8.10.2  Non-trivial convergence   ntrivcvgap 11540
                  4.8.10.3  Complex products   cprod 11542
                  4.8.10.4  Finite products   fprodseq 11575
      ​4.9  Elementary trigonometry
            4.9.1  The exponential, sine, and cosine functions   ce 11634
                  4.9.1.1  The circle constant (tau = 2 pi)   ctau 11766
            4.9.2  _e is irrational   eirraplem 11768
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11778
            ​*5.1.2  Even and odd numbers   evenelz 11855
            5.1.3  The division algorithm   divalglemnn 11906
            5.1.4  The greatest common divisor operator   cgcd 11926
            5.1.5  Bézout's identity   bezoutlemnewy 11980
            5.1.6  Decidable sets of integers   nnmindc 12018
            5.1.7  Algorithms   nn0seqcvgd 12024
            5.1.8  Euclid's Algorithm   eucalgval2 12036
            ​*5.1.9  The least common multiple   clcm 12043
            ​*5.1.10  Coprimality and Euclid's lemma   coprmgcdb 12071
            5.1.11  Cancellability of congruences   congr 12083
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12090
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12127
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12144
            5.2.4  Properties of the canonical representation of a rational   cnumer 12164
            5.2.5  Euler's theorem   codz 12191
            5.2.6  Arithmetic modulo a prime number   modprm1div 12230
            5.2.7  Pythagorean Triples   coprimeprodsq 12240
            5.2.8  The prime count function   cpc 12267
            5.2.9  Pocklington's theorem   prmpwdvds 12336
            5.2.10  Infinite primes theorem   infpnlem1 12340
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12344
            5.2.12  Lagrange's four-square theorem   cgz 12350
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12376
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12441
            6.1.2  Slot definitions   cplusg 12518
            6.1.3  Definition of the structure product   crest 12636
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 12664
            ​*7.1.2  Identity elements   mgmidmo 12683
            ​*7.1.3  Semigroups   csgrp 12699
            ​*7.1.4  Definition and basic properties of monoids   cmnd 12709
            7.1.5  Monoid homomorphisms and submonoids   cmhm 12739
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 12767
            ​*7.2.2  Group multiple operation   cmg 12872
            7.2.3  Abelian groups   ccmn 12915
                  7.2.3.1  Definition and basic properties   ccmn 12915
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 12957
            ​*7.3.2  Ring unity (multiplicative identity)   cur 12968
            7.3.3  Semirings   csrg 12972
            7.3.4  Definition and basic properties of unital rings   crg 13005
            7.3.5  Opposite ring   coppr 13064
            7.3.6  Divisibility   cdsr 13080
            7.3.7  Ring homomorphisms   crh 13142
      ​7.4  The complex numbers as an algebraic extensible structure
            7.4.1  Definition and basic properties   cpsmet 13146
​PART 8  BASIC TOPOLOGY
      ​8.1  Topology
            ​*8.1.1  Topological spaces   ctop 13162
                  8.1.1.1  Topologies   ctop 13162
                  8.1.1.2  Topologies on sets   ctopon 13175
                  8.1.1.3  Topological spaces   ctps 13195
            8.1.2  Topological bases   ctb 13207
            8.1.3  Examples of topologies   distop 13252
            8.1.4  Closure and interior   ccld 13259
            8.1.5  Neighborhoods   cnei 13305
            8.1.6  Subspace topologies   restrcl 13334
            8.1.7  Limits and continuity in topological spaces   ccn 13352
            8.1.8  Product topologies   ctx 13419
            8.1.9  Continuous function-builders   cnmptid 13448
            8.1.10  Homeomorphisms   chmeo 13467
      ​8.2  Metric spaces
            8.2.1  Pseudometric spaces   psmetrel 13489
            8.2.2  Basic metric space properties   cxms 13503
            8.2.3  Metric space balls   blfvalps 13552
            8.2.4  Open sets of a metric space   mopnrel 13608
            8.2.5  Continuity in metric spaces   metcnp3 13678
            8.2.6  Topology on the reals   qtopbasss 13688
            8.2.7  Topological definitions using the reals   ccncf 13724
​PART 9  BASIC REAL AND COMPLEX ANALYSIS
            9.0.1  Dedekind cuts   dedekindeulemuub 13762
            9.0.2  Intermediate value theorem   ivthinclemlm 13779
      ​9.1  Derivatives
            9.1.1  Real and complex differentiation   climc 13790
                  9.1.1.1  Derivatives of functions of one complex or real variable   climc 13790
​PART 10  BASIC REAL AND COMPLEX FUNCTIONS
      ​10.1  Basic trigonometry
            10.1.1  The exponential, sine, and cosine functions (cont.)   efcn 13856
            10.1.2  Properties of pi = 3.14159...   pilem1 13867
            10.1.3  The natural logarithm on complex numbers   clog 13944
            ​*10.1.4  Logarithms to an arbitrary base   clogb 14028
            10.1.5  Quartic binomial expansion   binom4 14064
      ​10.2  Basic number theory
            ​*10.2.1  Quadratic residues and the Legendre symbol   clgs 14065
            10.2.2  All primes 4n+1 are the sum of two squares   2sqlem1 14117
​PART 11  GUIDES AND MISCELLANEA
      ​11.1  Guides (conventions, explanations, and examples)
            ​*11.1.1  Conventions   conventions 14129
            11.1.2  Definitional examples   ex-or 14130
​PART 12  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​12.1  Mathboxes for user contributions
            12.1.1  Mathbox guidelines   mathbox 14140
      ​12.2  Mathbox for BJ
            12.2.1  Propositional calculus   bj-nnsn 14141
                  ​*12.2.1.1  Stable formulas   bj-trst 14147
                  12.2.1.2  Decidable formulas   bj-trdc 14160
            12.2.2  Predicate calculus   bj-ex 14170
            12.2.3  Set theorey miscellaneous   bj-el2oss1o 14182
            ​*12.2.4  Extensionality   bj-vtoclgft 14183
            ​*12.2.5  Decidability of classes   wdcin 14201
            12.2.6  Disjoint union   djucllem 14208
            12.2.7  Miscellaneous   funmptd 14211
            ​*12.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 14220
                  *12.2.8.1  Bounded formulas   wbd 14220
                  ​*12.2.8.2  Bounded classes   wbdc 14248
            ​*12.2.9  CZF: Bounded separation   ax-bdsep 14292
                  12.2.9.1  Delta_0-classical logic   ax-bj-d0cl 14332
                  12.2.9.2  Inductive classes and the class of natural number ordinals   wind 14334
                  ​*12.2.9.3  The first three Peano postulates   bj-peano2 14347
            ​*12.2.10  CZF: Infinity   ax-infvn 14349
                  *12.2.10.1  The set of natural number ordinals   ax-infvn 14349
                  ​*12.2.10.2  Peano's fifth postulate   bdpeano5 14351
                  ​*12.2.10.3  Bounded induction and Peano's fourth postulate   findset 14353
            ​*12.2.11  CZF: Set induction   setindft 14373
                  *12.2.11.1  Set induction   setindft 14373
                  ​*12.2.11.2  Full induction   bj-findis 14387
            ​*12.2.12  CZF: Strong collection   ax-strcoll 14390
            ​*12.2.13  CZF: Subset collection   ax-sscoll 14395
            12.2.14  Real numbers   ax-ddkcomp 14397
      ​12.3  Mathbox for Jim Kingdon
            12.3.1  Propositional and predicate logic   nnnotnotr 14398
            12.3.2  Natural numbers   ss1oel2o 14399
            12.3.3  The power set of a singleton   pwtrufal 14403
            12.3.4  Omniscience of NN+oo   0nninf 14409
            12.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 14426
            12.3.6  Real and complex numbers   qdencn 14431
            ​*12.3.7  Analytic omniscience principles   trilpolemclim 14440
            12.3.8  Supremum and infimum   supfz 14472
            12.3.9  Circle constant   taupi 14474
      ​12.4  Mathbox for Mykola Mostovenko
      ​12.5  Mathbox for David A. Wheeler
            12.5.1  Testable propositions   dftest 14476
            ​*12.5.2  Allsome quantifier   walsi 14477