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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
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
      6.2  The complex numbers as an algebraic extensible structure
PART 7  BASIC TOPOLOGY
      7.1  Topology
      7.2  Metric spaces
PART 8  BASIC REAL AND COMPLEX ANALYSIS
      8.1  Derivatives
PART 9  BASIC REAL AND COMPLEX FUNCTIONS
      9.1  Basic trigonometry
PART 10  GUIDES AND MISCELLANEA
      10.1  Guides (conventions, explanations, and examples)
PART 11  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      11.1  Mathboxes for user contributions
      11.2  Mathbox for BJ
      11.3  Mathbox for Jim Kingdon
      11.4  Mathbox for Mykola Mostovenko
      11.5  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 103
            1.2.5  Logical negation (intuitionistic)   ax-in1 588
            1.2.6  Logical disjunction   wo 682
            1.2.7  Stable propositions   wstab 800
            1.2.8  Decidable propositions   wdc 804
            *1.2.9  Theorems of decidable propositions   const 822
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 886
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 946
            1.2.12  True and false constants   wal 1314
                  *1.2.12.1  Universal quantifier for use by df-tru   wal 1314
                  *1.2.12.2  Equality predicate for use by df-tru   cv 1315
                  1.2.12.3  Define the true and false constants   wtru 1317
            1.2.13  Logical 'xor'   wxo 1338
            *1.2.14  Truth tables: Operations on true and false constants   truantru 1364
            *1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1386
            1.2.16  Logical implication (continued)   syl6an 1395
      1.3  Predicate calculus mostly without distinct variables
            *1.3.1  Universal quantifier (continued)   ax-5 1408
            *1.3.2  Equality predicate (continued)   weq 1464
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1491
            1.3.4  Introduce Axiom of Existence   ax-i9 1495
            1.3.5  Additional intuitionistic axioms   ax-ial 1499
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1501
            1.3.7  The existential quantifier   19.8a 1554
            1.3.8  Equality theorems without distinct variables   a9e 1659
            1.3.9  Axioms ax-10 and ax-11   ax10o 1678
            1.3.10  Substitution (without distinct variables)   wsb 1720
            1.3.11  Theorems using axiom ax-11   equs5a 1750
      1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1767
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1778
            1.4.3  More theorems related to ax-11 and substitution   albidv 1780
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1814
            1.4.5  More substitution theorems   hbs1 1891
            1.4.6  Existential uniqueness   weu 1977
            *1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2075
*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 2099
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2103
            2.1.3  Class form not-free predicate   wnfc 2245
            2.1.4  Negated equality and membership   wne 2285
                  2.1.4.1  Negated equality   wne 2285
                  2.1.4.2  Negated membership   wnel 2380
            2.1.5  Restricted quantification   wral 2393
            2.1.6  The universal class   cvv 2660
            *2.1.7  Conditional equality (experimental)   wcdeq 2865
            2.1.8  Russell's Paradox   ru 2881
            2.1.9  Proper substitution of classes for sets   wsbc 2882
            2.1.10  Proper substitution of classes for sets into classes   csb 2975
            2.1.11  Define basic set operations and relations   cdif 3038
            2.1.12  Subclasses and subsets   df-ss 3054
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3156
                  2.1.13.1  The difference of two classes   dfdif3 3156
                  2.1.13.2  The union of two classes   elun 3187
                  2.1.13.3  The intersection of two classes   elin 3229
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3277
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3312
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3326
            2.1.14  The empty set   c0 3333
            2.1.15  Conditional operator   cif 3444
            2.1.16  Power classes   cpw 3480
            2.1.17  Unordered and ordered pairs   csn 3497
            2.1.18  The union of a class   cuni 3706
            2.1.19  The intersection of a class   cint 3741
            2.1.20  Indexed union and intersection   ciun 3783
            2.1.21  Disjointness   wdisj 3876
            2.1.22  Binary relations   wbr 3899
            2.1.23  Ordered-pair class abstractions (class builders)   copab 3958
            2.1.24  Transitive classes   wtr 3996
      2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4013
            2.2.2  Introduce the Axiom of Separation   ax-sep 4016
            2.2.3  Derive the Null Set Axiom   zfnuleu 4022
            2.2.4  Theorems requiring subset and intersection existence   nalset 4028
            2.2.5  Theorems requiring empty set existence   class2seteq 4057
            2.2.6  Collection principle   bnd 4066
      2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4068
            2.3.2  A notation for excluded middle   wem 4088
            2.3.3  Axiom of Pairing   ax-pr 4101
            2.3.4  Ordered pair theorem   opm 4126
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4149
            2.3.6  Power class of union and intersection   pwin 4174
            2.3.7  Epsilon and identity relations   cep 4179
            2.3.8  Partial and complete ordering   wpo 4186
            2.3.9  Founded and set-like relations   wfrfor 4219
            2.3.10  Ordinals   word 4254
      2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4325
            2.4.2  Ordinals (continued)   ordon 4372
      2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4417
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4422
            2.5.3  Transfinite induction   tfi 4466
      2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4472
            2.6.2  The natural numbers (i.e. finite ordinals)   com 4474
            2.6.3  Peano's postulates   peano1 4478
            2.6.4  Finite induction (for finite ordinals)   find 4483
            2.6.5  The Natural Numbers (continued)   nn0suc 4488
            2.6.6  Relations   cxp 4507
            2.6.7  Definite description binder (inverted iota)   cio 5056
            2.6.8  Functions   wfun 5087
            2.6.9  Restricted iota (description binder)   crio 5697
            2.6.10  Operations   co 5742
            2.6.11  Maps-to notation   elmpocl 5936
            2.6.12  Function operation   cof 5948
            2.6.13  Functions (continued)   resfunexgALT 5976
            2.6.14  First and second members of an ordered pair   c1st 6004
            *2.6.15  Special maps-to operations   opeliunxp2f 6103
            2.6.16  Function transposition   ctpos 6109
            2.6.17  Undefined values   pwuninel2 6147
            2.6.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6149
            2.6.19  "Strong" transfinite recursion   crecs 6169
            2.6.20  Recursive definition generator   crdg 6234
            2.6.21  Finite recursion   cfrec 6255
            2.6.22  Ordinal arithmetic   c1o 6274
            2.6.23  Natural number arithmetic   nna0 6338
            2.6.24  Equivalence relations and classes   wer 6394
            2.6.25  The mapping operation   cmap 6510
            2.6.26  Infinite Cartesian products   cixp 6560
            2.6.27  Equinumerosity   cen 6600
            2.6.28  Equinumerosity (cont.)   xpf1o 6706
            2.6.29  Pigeonhole Principle   phplem1 6714
            2.6.30  Finite sets   fict 6730
            2.6.31  Schroeder-Bernstein Theorem   sbthlem1 6813
            2.6.32  Finite intersections   cfi 6824
            2.6.33  Supremum and infimum   csup 6837
            2.6.34  Ordinal isomorphism   ordiso2 6888
            2.6.35  Disjoint union   cdju 6890
                  2.6.35.1  Disjoint union   cdju 6890
                  *2.6.35.2  Left and right injections of a disjoint union   cinl 6898
                  2.6.35.3  Universal property of the disjoint union   djuss 6923
                  2.6.35.4  Dominance and equinumerosity properties of disjoint union   djudom 6946
                  2.6.35.5  Older definition temporarily kept for comparison, to be deleted   cdjud 6955
                  2.6.35.6  Countable sets   0ct 6960
            2.6.36  Omniscient sets   comni 6972
            2.6.37  Markov's principle   cmarkov 6993
            2.6.38  Cardinal numbers   ccrd 7003
            2.6.39  Axiom of Choice equivalents   wac 7029
            2.6.40  Cardinal number arithmetic   endjudisj 7034
*PART 3  CHOICE PRINCIPLES
      3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7045
*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 7048
            4.1.2  Final derivation of real and complex number postulates   axcnex 7635
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7679
      4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 7712
            4.2.2  Infinity and the extended real number system   cpnf 7765
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 7799
            4.2.4  Ordering on reals   lttr 7806
            4.2.5  Initial properties of the complex numbers   mul12 7859
      4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 7888
            4.3.2  Subtraction   cmin 7901
            4.3.3  Multiplication   kcnktkm1cn 8113
            4.3.4  Ordering on reals (cont.)   ltadd2 8149
            4.3.5  Real Apartness   creap 8303
            4.3.6  Complex Apartness   cap 8310
            4.3.7  Reciprocals   recextlem1 8379
            4.3.8  Division   cdiv 8399
            4.3.9  Ordering on reals (cont.)   ltp1 8566
            4.3.10  Suprema   lbreu 8667
            4.3.11  Imaginary and complex number properties   crap0 8680
      4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 8684
            4.4.2  Principle of mathematical induction   nnind 8700
            *4.4.3  Decimal representation of numbers   c2 8735
            *4.4.4  Some properties of specific numbers   neg1cn 8789
            4.4.5  Simple number properties   halfcl 8904
            4.4.6  The Archimedean property   arch 8932
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 8935
            *4.4.8  Extended nonnegative integers   cxnn0 8998
            4.4.9  Integers (as a subset of complex numbers)   cz 9012
            4.4.10  Decimal arithmetic   cdc 9140
            4.4.11  Upper sets of integers   cuz 9282
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9367
            4.4.13  Complex numbers as pairs of reals   cnref1o 9396
      4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9397
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9511
            4.5.3  Real number intervals   cioo 9626
            4.5.4  Finite intervals of integers   cfz 9745
            *4.5.5  Finite intervals of nonnegative integers   elfz2nn0 9847
            4.5.6  Half-open integer ranges   cfzo 9874
            4.5.7  Rational numbers (cont.)   qtri3or 9975
      4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 9996
            4.6.2  The modulo (remainder) operation   cmo 10050
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10127
            4.6.4  Strong induction over upper sets of integers   uzsinds 10170
            4.6.5  The infinite sequence builder "seq"   cseq 10173
            4.6.6  Integer powers   cexp 10247
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10420
            4.6.8  Factorial function   cfa 10426
            4.6.9  The binomial coefficient operation   cbc 10448
            4.6.10  The ` # ` (set size) function   chash 10476
      4.7  Elementary real and complex functions
            4.7.1  The "shift" operation   cshi 10541
            4.7.2  Real and imaginary parts; conjugate   ccj 10566
            4.7.3  Sequence convergence   caucvgrelemrec 10706
            4.7.4  Square root; absolute value   csqrt 10723
            4.7.5  The maximum of two real numbers   maxcom 10930
            4.7.6  The minimum of two real numbers   mincom 10955
            4.7.7  The maximum of two extended reals   xrmaxleim 10968
            4.7.8  The minimum of two extended reals   xrnegiso 10986
      4.8  Elementary limits and convergence
            4.8.1  Limits   cli 11002
            4.8.2  Finite and infinite sums   csu 11077
            4.8.3  The binomial theorem   binomlem 11207
            4.8.4  Infinite sums (cont.)   isumshft 11214
            4.8.5  Miscellaneous converging and diverging sequences   divcnv 11221
            4.8.6  Arithmetic series   arisum 11222
            4.8.7  Geometric series   expcnvap0 11226
            4.8.8  Ratio test for infinite series convergence   cvgratnnlembern 11247
            4.8.9  Mertens' theorem   mertenslemub 11258
      4.9  Elementary trigonometry
            4.9.1  The exponential, sine, and cosine functions   ce 11262
            4.9.2  _e is irrational   eirraplem 11395
*PART 5  ELEMENTARY NUMBER THEORY
      5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11405
            *5.1.2  Even and odd numbers   evenelz 11476
            5.1.3  The division algorithm   divalglemnn 11527
            5.1.4  The greatest common divisor operator   cgcd 11547
            5.1.5  Bézout's identity   bezoutlemnewy 11596
            5.1.6  Algorithms   nn0seqcvgd 11634
            5.1.7  Euclid's Algorithm   eucalgval2 11646
            *5.1.8  The least common multiple   clcm 11653
            *5.1.9  Coprimality and Euclid's lemma   coprmgcdb 11681
            5.1.10  Cancellability of congruences   congr 11693
      5.2  Elementary prime number theory
            *5.2.1  Elementary properties   cprime 11700
            *5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 11734
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 11751
            5.2.4  Properties of the canonical representation of a rational   cnumer 11770
            5.2.5  Euler's theorem   cphi 11797
      5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 11816
PART 6  BASIC STRUCTURES
      6.1  Extensible structures
            *6.1.1  Basic definitions   cstr 11866
            6.1.2  Slot definitions   cplusg 11932
            6.1.3  Definition of the structure product   crest 12031
      6.2  The complex numbers as an algebraic extensible structure
            6.2.1  Definition and basic properties   cpsmet 12059
PART 7  BASIC TOPOLOGY
      7.1  Topology
            *7.1.1  Topological spaces   ctop 12075
                  7.1.1.1  Topologies   ctop 12075
                  7.1.1.2  Topologies on sets   ctopon 12088
                  7.1.1.3  Topological spaces   ctps 12108
            7.1.2  Topological bases   ctb 12120
            7.1.3  Examples of topologies   distop 12165
            7.1.4  Closure and interior   ccld 12172
            7.1.5  Neighborhoods   cnei 12218
            7.1.6  Subspace topologies   restrcl 12247
            7.1.7  Limits and continuity in topological spaces   ccn 12265
            7.1.8  Product topologies   ctx 12332
            7.1.9  Continuous function-builders   cnmptid 12361
            7.1.10  Homeomorphisms   chmeo 12380
      7.2  Metric spaces
            7.2.1  Pseudometric spaces   psmetrel 12402
            7.2.2  Basic metric space properties   cxms 12416
            7.2.3  Metric space balls   blfvalps 12465
            7.2.4  Open sets of a metric space   mopnrel 12521
            7.2.5  Continuity in metric spaces   metcnp3 12591
            7.2.6  Topology on the reals   qtopbasss 12601
            7.2.7  Topological definitions using the reals   ccncf 12637
PART 8  BASIC REAL AND COMPLEX ANALYSIS
            8.0.1  Dedekind cuts   dedekindeulemuub 12675
            8.0.2  Intermediate value theorem   ivthinclemlm 12692
      8.1  Derivatives
            8.1.1  Real and complex differentiation   climc 12703
                  8.1.1.1  Derivatives of functions of one complex or real variable   climc 12703
PART 9  BASIC REAL AND COMPLEX FUNCTIONS
      9.1  Basic trigonometry
            9.1.1  The exponential, sine, and cosine functions (cont.)   efcn 12768
            9.1.2  Properties of pi = 3.14159...   pilem1 12771
PART 10  GUIDES AND MISCELLANEA
      10.1  Guides (conventions, explanations, and examples)
            *10.1.1  Conventions   conventions 12829
            10.1.2  Definitional examples   ex-or 12830
PART 11  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      11.1  Mathboxes for user contributions
            11.1.1  Mathbox guidelines   mathbox 12840
      11.2  Mathbox for BJ
            11.2.1  Propositional calculus   bj-nnsn 12841
                  11.2.1.1  Stable formulas   bj-trst 12847
                  11.2.1.2  Decidable formulas   bj-trdc 12855
            11.2.2  Predicate calculus   bj-ex 12865
            11.2.3  Set theorey miscellaneous   bj-el2oss1o 12877
            *11.2.4  Extensionality   bj-vtoclgft 12878
            *11.2.5  Decidability of classes   wdcin 12896
            11.2.6  Disjoint union   djucllem 12903
            *11.2.7  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 12906
                  *11.2.7.1  Bounded formulas   wbd 12906
                  *11.2.7.2  Bounded classes   wbdc 12934
            *11.2.8  CZF: Bounded separation   ax-bdsep 12978
                  11.2.8.1  Delta_0-classical logic   ax-bj-d0cl 13018
                  11.2.8.2  Inductive classes and the class of natural numbers (finite ordinals)   wind 13020
                  *11.2.8.3  The first three Peano postulates   bj-peano2 13033
            *11.2.9  CZF: Infinity   ax-infvn 13035
                  *11.2.9.1  The set of natural numbers (finite ordinals)   ax-infvn 13035
                  *11.2.9.2  Peano's fifth postulate   bdpeano5 13037
                  *11.2.9.3  Bounded induction and Peano's fourth postulate   findset 13039
            *11.2.10  CZF: Set induction   setindft 13059
                  *11.2.10.1  Set induction   setindft 13059
                  *11.2.10.2  Full induction   bj-findis 13073
            *11.2.11  CZF: Strong collection   ax-strcoll 13076
            *11.2.12  CZF: Subset collection   ax-sscoll 13081
            11.2.13  Real numbers   ax-ddkcomp 13083
      11.3  Mathbox for Jim Kingdon
            11.3.1  Natural numbers   el2oss1o 13084
            11.3.2  The power set of a singleton   pwtrufal 13088
            11.3.3  Omniscience of NN+oo   0nninf 13093
            11.3.4  Schroeder-Bernstein Theorem   exmidsbthrlem 13113
            11.3.5  Real and complex numbers   qdencn 13118
            11.3.6  Supremum and infimum   supfz 13133
      11.4  Mathbox for Mykola Mostovenko
      11.5  Mathbox for David A. Wheeler
            11.5.1  Testable propositions   dftest 13136
            *11.5.2  Allsome quantifier   walsi 13137

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