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Table of Contents Summary
PART 1  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  REAL AND COMPLEX NUMBERS
      3.1  Construction and axiomatization of real and complex numbers
      3.2  Derive the basic properties from the field axioms
      3.3  Real and complex numbers - basic operations
      3.4  Integer sets
      3.5  Order sets
      3.6  Elementary integer functions
      3.7  Elementary real and complex functions
      3.8  Elementary limits and convergence
PART 4  ELEMENTARY NUMBER THEORY
      4.1  Elementary properties of divisibility
      4.2  Elementary prime number theory
      4.3  Cardinality of real and complex number subsets
PART 5  GUIDES AND MISCELLANEA
      5.1  Guides (conventions, explanations, and examples)
PART 6  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      6.1  Mathboxes for user contributions
      6.2  Mathbox for BJ
      6.3  Mathbox for Jim Kingdon
      6.4  Mathbox for Mykola Mostovenko
      6.5  Mathbox for David A. Wheeler

Detailed Table of Contents
(* means the section header has a description)
PART 1  FIRST ORDER LOGIC WITH EQUALITY
      *1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   a1ii 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-1 5
            *1.2.3  Logical implication   mp2b 8
            1.2.4  Logical conjunction and logical equivalence   wa 102
            1.2.5  Logical negation (intuitionistic)   ax-in1 577
            1.2.6  Logical disjunction   wo 662
            1.2.7  Stable propositions   wstab 773
            1.2.8  Decidable propositions   wdc 776
            *1.2.9  Theorems of decidable propositions   condc 783
            1.2.10  Testable propositions   dftest 856
            1.2.11  Miscellaneous theorems of propositional calculus   pm5.21nd 859
            1.2.12  Abbreviated conjunction and disjunction of three wff's   w3o 919
            1.2.13  True and false constants   wal 1283
                  *1.2.13.1  Universal quantifier for use by df-tru   wal 1283
                  *1.2.13.2  Equality predicate for use by df-tru   cv 1284
                  1.2.13.3  Define the true and false constants   wtru 1286
            1.2.14  Logical 'xor'   wxo 1307
            *1.2.15  Truth tables: Operations on true and false constants   truantru 1333
            *1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1355
            1.2.17  Logical implication (continued)   syl6an 1364
      1.3  Predicate calculus mostly without distinct variables
            *1.3.1  Universal quantifier (continued)   ax-5 1377
            *1.3.2  Equality predicate (continued)   weq 1433
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1460
            1.3.4  Introduce Axiom of Existence   ax-i9 1464
            1.3.5  Additional intuitionistic axioms   ax-ial 1468
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1470
            1.3.7  The existential quantifier   19.8a 1523
            1.3.8  Equality theorems without distinct variables   a9e 1627
            1.3.9  Axioms ax-10 and ax-11   ax10o 1645
            1.3.10  Substitution (without distinct variables)   wsb 1687
            1.3.11  Theorems using axiom ax-11   equs5a 1717
      1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1734
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1745
            1.4.3  More theorems related to ax-11 and substitution   albidv 1747
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1781
            1.4.5  More substitution theorems   hbs1 1857
            1.4.6  Existential uniqueness   weu 1943
            *1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2041
*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 2065
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2069
            2.1.3  Class form not-free predicate   wnfc 2210
            2.1.4  Negated equality and membership   wne 2249
                  2.1.4.1  Negated equality   wne 2249
                  2.1.4.2  Negated membership   wnel 2344
            2.1.5  Restricted quantification   wral 2353
            2.1.6  The universal class   cvv 2612
            *2.1.7  Conditional equality (experimental)   wcdeq 2809
            2.1.8  Russell's Paradox   ru 2825
            2.1.9  Proper substitution of classes for sets   wsbc 2826
            2.1.10  Proper substitution of classes for sets into classes   csb 2919
            2.1.11  Define basic set operations and relations   cdif 2981
            2.1.12  Subclasses and subsets   df-ss 2997
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3094
                  2.1.13.1  The difference of two classes   dfdif3 3094
                  2.1.13.2  The union of two classes   elun 3125
                  2.1.13.3  The intersection of two classes   elin 3167
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3214
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3248
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3262
            2.1.14  The empty set   c0 3269
            2.1.15  Conditional operator   cif 3373
            2.1.16  Power classes   cpw 3406
            2.1.17  Unordered and ordered pairs   csn 3422
            2.1.18  The union of a class   cuni 3627
            2.1.19  The intersection of a class   cint 3662
            2.1.20  Indexed union and intersection   ciun 3704
            2.1.21  Disjointness   wdisj 3792
            2.1.22  Binary relations   wbr 3811
            2.1.23  Ordered-pair class abstractions (class builders)   copab 3864
            2.1.24  Transitive classes   wtr 3901
      2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 3919
            2.2.2  Introduce the Axiom of Separation   ax-sep 3922
            2.2.3  Derive the Null Set Axiom   zfnuleu 3928
            2.2.4  Theorems requiring subset and intersection existence   nalset 3934
            2.2.5  Theorems requiring empty set existence   class2seteq 3963
            2.2.6  Collection principle   bnd 3972
      2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 3974
            2.3.2  A notation for excluded middle   wem 3992
            2.3.3  Axiom of Pairing   ax-pr 3999
            2.3.4  Ordered pair theorem   opm 4024
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4047
            2.3.6  Power class of union and intersection   pwin 4072
            2.3.7  Epsilon and identity relations   cep 4077
            2.3.8  Partial and complete ordering   wpo 4084
            2.3.9  Founded and set-like relations   wfrfor 4117
            2.3.10  Ordinals   word 4152
      2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4223
            2.4.2  Ordinals (continued)   ordon 4265
      2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4310
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4315
            2.5.3  Transfinite induction   tfi 4359
      2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4365
            2.6.2  The natural numbers (i.e. finite ordinals)   com 4367
            2.6.3  Peano's postulates   peano1 4371
            2.6.4  Finite induction (for finite ordinals)   find 4376
            2.6.5  The Natural Numbers (continued)   nn0suc 4381
            2.6.6  Relations   cxp 4398
            2.6.7  Definite description binder (inverted iota)   cio 4931
            2.6.8  Functions   wfun 4962
            2.6.9  Restricted iota (description binder)   crio 5545
            2.6.10  Operations   co 5590
            2.6.11  "Maps to" notation   elmpt2cl 5776
            2.6.12  Function operation   cof 5788
            2.6.13  Functions (continued)   resfunexgALT 5815
            2.6.14  First and second members of an ordered pair   c1st 5843
            *2.6.15  Special "Maps to" operations   mpt2xopn0yelv 5935
            2.6.16  Function transposition   ctpos 5940
            2.6.17  Undefined values   pwuninel2 5978
            2.6.18  Functions on ordinals; strictly monotone ordinal functions   iunon 5980
            2.6.19  "Strong" transfinite recursion   crecs 6000
            2.6.20  Recursive definition generator   crdg 6065
            2.6.21  Finite recursion   cfrec 6086
            2.6.22  Ordinal arithmetic   c1o 6105
            2.6.23  Natural number arithmetic   nna0 6166
            2.6.24  Equivalence relations and classes   wer 6218
            2.6.25  The mapping operation   cmap 6334
            2.6.26  Equinumerosity   cen 6384
            2.6.27  Equinumerosity (cont.)   xpf1o 6489
            2.6.28  Pigeonhole Principle   phplem1 6497
            2.6.29  Finite sets   fict 6513
            2.6.30  Supremum and infimum   csup 6583
            2.6.31  Ordinal isomorphism   ordiso2 6634
            2.6.32  Disjoint union   cdju 6636
                  2.6.32.1  Disjoint union   cdju 6636
                  *2.6.32.2  Left and right injections of a disjoint union   cinl 6643
                  2.6.32.3  Universal property of the disjoint union   djuss 6667
                  2.6.32.4  Older definition temporarily kept for comparison, to be deleted   cdjud 6688
            2.6.33  Omniscient sets   comni 6694
            2.6.34  Cardinal numbers   ccrd 6709
*PART 3  REAL AND COMPLEX NUMBERS
      3.1  Construction and axiomatization of real and complex numbers
            3.1.1  Dedekind-cut construction of real and complex numbers   cnpi 6733
            3.1.2  Final derivation of real and complex number postulates   axcnex 7298
            3.1.3  Real and complex number postulates restated as axioms   ax-cnex 7338
      3.2  Derive the basic properties from the field axioms
            3.2.1  Some deductions from the field axioms for complex numbers   cnex 7368
            3.2.2  Infinity and the extended real number system   cpnf 7421
            3.2.3  Restate the ordering postulates with extended real "less than"   axltirr 7455
            3.2.4  Ordering on reals   lttr 7461
            3.2.5  Initial properties of the complex numbers   mul12 7513
      3.3  Real and complex numbers - basic operations
            3.3.1  Addition   add12 7542
            3.3.2  Subtraction   cmin 7555
            3.3.3  Multiplication   kcnktkm1cn 7763
            3.3.4  Ordering on reals (cont.)   ltadd2 7799
            3.3.5  Real Apartness   creap 7950
            3.3.6  Complex Apartness   cap 7957
            3.3.7  Reciprocals   recextlem1 8017
            3.3.8  Division   cdiv 8036
            3.3.9  Ordering on reals (cont.)   ltp1 8198
            3.3.10  Suprema   lbreu 8299
            3.3.11  Imaginary and complex number properties   crap0 8311
      3.4  Integer sets
            3.4.1  Positive integers (as a subset of complex numbers)   cn 8315
            3.4.2  Principle of mathematical induction   nnind 8331
            *3.4.3  Decimal representation of numbers   c2 8365
            *3.4.4  Some properties of specific numbers   neg1cn 8420
            3.4.5  Simple number properties   halfcl 8533
            3.4.6  The Archimedean property   arch 8561
            3.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 8564
            *3.4.8  Extended nonnegative integers   cxnn0 8627
            3.4.9  Integers (as a subset of complex numbers)   cz 8645
            3.4.10  Decimal arithmetic   cdc 8771
            3.4.11  Upper sets of integers   cuz 8913
            3.4.12  Rational numbers (as a subset of complex numbers)   cq 8998
            3.4.13  Complex numbers as pairs of reals   cnref1o 9027
      3.5  Order sets
            3.5.1  Positive reals (as a subset of complex numbers)   crp 9028
            3.5.2  Infinity and the extended real number system (cont.)   cxne 9134
            3.5.3  Real number intervals   cioo 9200
            3.5.4  Finite intervals of integers   cfz 9318
            *3.5.5  Finite intervals of nonnegative integers   elfz2nn0 9418
            3.5.6  Half-open integer ranges   cfzo 9442
            3.5.7  Rational numbers (cont.)   qtri3or 9542
      3.6  Elementary integer functions
            3.6.1  The floor and ceiling functions   cfl 9563
            3.6.2  The modulo (remainder) operation   cmo 9617
            3.6.3  Miscellaneous theorems about integers   frec2uz0d 9694
            3.6.4  Strong induction over upper sets of integers   uzsinds 9736
            3.6.5  The infinite sequence builder "seq"   cseq 9739
            3.6.6  Integer powers   cexp 9790
            3.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 9961
            3.6.8  Factorial function   cfa 9967
            3.6.9  The binomial coefficient operation   cbc 9989
            3.6.10  The ` # ` (set size) function   chash 10017
      3.7  Elementary real and complex functions
            3.7.1  The "shift" operation   cshi 10075
            3.7.2  Real and imaginary parts; conjugate   ccj 10099
            3.7.3  Sequence convergence   caucvgrelemrec 10238
            3.7.4  Square root; absolute value   csqrt 10255
            3.7.5  The maximum of two real numbers   maxcom 10462
            3.7.6  The minimum of two real numbers   mincom 10484
      3.8  Elementary limits and convergence
            3.8.1  Limits   cli 10490
            3.8.2  Finite and infinite sums   csu 10563
*PART 4  ELEMENTARY NUMBER THEORY
      4.1  Elementary properties of divisibility
            4.1.1  The divides relation   cdvds 10575
            *4.1.2  Even and odd numbers   evenelz 10646
            4.1.3  The division algorithm   divalglemnn 10697
            4.1.4  The greatest common divisor operator   cgcd 10717
            4.1.5  Bézout's identity   bezoutlemnewy 10764
            4.1.6  Algorithms   nn0seqcvgd 10802
            4.1.7  Euclid's Algorithm   eucalgval2 10814
            *4.1.8  The least common multiple   clcm 10821
            *4.1.9  Coprimality and Euclid's lemma   coprmgcdb 10849
            4.1.10  Cancellability of congruences   congr 10861
      4.2  Elementary prime number theory
            *4.2.1  Elementary properties   cprime 10868
            *4.2.2  Coprimality and Euclid's lemma (cont.)   coprm 10902
            4.2.3  Non-rationality of square root of 2   sqrt2irrlem 10919
            4.2.4  Properties of the canonical representation of a rational   cnumer 10938
            4.2.5  Euler's theorem   cphi 10965
      4.3  Cardinality of real and complex number subsets
            4.3.1  Countability of integers and rationals   oddennn 10984
PART 5  GUIDES AND MISCELLANEA
      5.1  Guides (conventions, explanations, and examples)
            *5.1.1  Conventions   conventions 10991
            5.1.2  Definitional examples   ex-or 10992
PART 6  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      6.1  Mathboxes for user contributions
            6.1.1  Mathbox guidelines   mathbox 11001
      6.2  Mathbox for BJ
            6.2.1  Propositional calculus   nnexmid 11002
            6.2.2  Predicate calculus   bj-ex 11005
            *6.2.3  Extensionality   bj-vtoclgft 11017
            *6.2.4  Dedidability of classes   wdcin 11035
            6.2.5  Disjoint union   djucllem 11042
            *6.2.6  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 11045
                  *6.2.6.1  Bounded formulas   wbd 11045
                  *6.2.6.2  Bounded classes   wbdc 11073
            *6.2.7  CZF: Bounded separation   ax-bdsep 11117
                  6.2.7.1  Delta_0-classical logic   ax-bj-d0cl 11157
                  6.2.7.2  Inductive classes and the class of natural numbers (finite ordinals)   wind 11163
                  *6.2.7.3  The first three Peano postulates   bj-peano2 11176
            *6.2.8  CZF: Infinity   ax-infvn 11178
                  *6.2.8.1  The set of natural numbers (finite ordinals)   ax-infvn 11178
                  *6.2.8.2  Peano's fifth postulate   bdpeano5 11180
                  *6.2.8.3  Bounded induction and Peano's fourth postulate   findset 11182
            *6.2.9  CZF: Set induction   setindft 11202
                  *6.2.9.1  Set induction   setindft 11202
                  *6.2.9.2  Full induction   bj-findis 11216
            *6.2.10  CZF: Strong collection   ax-strcoll 11219
            *6.2.11  CZF: Subset collection   ax-sscoll 11224
            6.2.12  Real numbers   ax-ddkcomp 11226
      6.3  Mathbox for Jim Kingdon
      6.4  Mathbox for Mykola Mostovenko
      6.5  Mathbox for David A. Wheeler
            *6.5.1  Allsome quantifier   walsi 11229

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