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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
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 Mykola Mostovenko
6.3  Mathbox for BJ
6.4  Mathbox for David A. Wheeler
6.5  Mathbox for Jim Kingdon

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 97
1.2.5  Logical negation (intuitionistic)   ax-in1 544
1.2.6  Logical disjunction   wo 629
1.2.7  Stable propositions   wstab 739
1.2.8  Decidable propositions   wdc 742
*1.2.9  Theorems of decidable propositions   condc 749
1.2.10  Testable propositions   dftest 822
1.2.11  Miscellaneous theorems of propositional calculus   pm5.21nd 825
1.2.12  Abbreviated conjunction and disjunction of three wff's   w3o 884
1.2.13  True and false constants   wal 1241
*1.2.13.1  Universal quantifier for use by df-tru   wal 1241
*1.2.13.2  Equality predicate for use by df-tru   cv 1242
1.2.13.3  Define the true and false constants   wtru 1244
1.2.14  Logical 'xor'   wxo 1266
*1.2.15  Truth tables: Operations on true and false constants   truantru 1292
*1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1314
1.2.17  Logical implication (continued)   syl6an 1323
1.3  Predicate calculus mostly without distinct variables
*1.3.1  Universal quantifier (continued)   ax-5 1336
*1.3.2  Equality predicate (continued)   weq 1392
1.3.3  Axiom ax-17 - first use of the \$d distinct variable statement   ax-17 1419
1.3.4  Introduce Axiom of Existence   ax-i9 1423
1.3.5  Additional intuitionistic axioms   ax-ial 1427
1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1429
1.3.7  The existential quantifier   19.8a 1482
1.3.8  Equality theorems without distinct variables   a9e 1586
1.3.9  Axioms ax-10 and ax-11   ax10o 1603
1.3.10  Substitution (without distinct variables)   wsb 1645
1.3.11  Theorems using axiom ax-11   equs5a 1675
1.4  Predicate calculus with distinct variables
1.4.1  Derive the axiom of distinct variables ax-16   spimv 1692
1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1703
1.4.3  More theorems related to ax-11 and substitution   albidv 1705
1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1738
1.4.5  More substitution theorems   hbs1 1814
1.4.6  Existential uniqueness   weu 1900
*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 1998
*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 2022
2.1.2  Class abstractions (a.k.a. class builders)   cab 2026
2.1.3  Class form not-free predicate   wnfc 2165
2.1.4  Negated equality and membership   wne 2204
2.1.4.1  Negated equality   neii 2208
2.1.4.2  Negated membership   neli 2299
2.1.5  Restricted quantification   wral 2306
2.1.6  The universal class   cvv 2557
*2.1.7  Conditional equality (experimental)   wcdeq 2747
2.1.8  Russell's Paradox   ru 2763
2.1.9  Proper substitution of classes for sets   wsbc 2764
2.1.10  Proper substitution of classes for sets into classes   csb 2852
2.1.11  Define basic set operations and relations   cdif 2914
2.1.12  Subclasses and subsets   df-ss 2931
2.1.13  The difference, union, and intersection of two classes   difeq1 3055
2.1.13.1  The difference of two classes   difeq1 3055
2.1.13.2  The union of two classes   elun 3084
2.1.13.3  The intersection of two classes   elin 3126
2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3167
2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3203
2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3217
2.1.14  The empty set   c0 3224
2.1.15  Conditional operator   cif 3331
2.1.16  Power classes   cpw 3359
2.1.17  Unordered and ordered pairs   csn 3375
2.1.18  The union of a class   cuni 3580
2.1.19  The intersection of a class   cint 3615
2.1.20  Indexed union and intersection   ciun 3657
2.1.21  Disjointness   wdisj 3745
2.1.22  Binary relations   wbr 3764
2.1.23  Ordered-pair class abstractions (class builders)   copab 3817
2.1.24  Transitive classes   wtr 3854
2.2  IZF Set Theory - add the Axioms of Collection and Separation
2.2.1  Introduce the Axiom of Collection   ax-coll 3872
2.2.2  Introduce the Axiom of Separation   ax-sep 3875
2.2.3  Derive the Null Set Axiom   zfnuleu 3881
2.2.4  Theorems requiring subset and intersection existence   nalset 3887
2.2.5  Theorems requiring empty set existence   class2seteq 3916
2.2.6  Collection principle   bnd 3925
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
2.3.1  Introduce the Axiom of Power Sets   ax-pow 3927
2.3.2  Axiom of Pairing   ax-pr 3944
2.3.3  Ordered pair theorem   opm 3971
2.3.4  Ordered-pair class abstractions (cont.)   opabid 3994
2.3.5  Power class of union and intersection   pwin 4019
2.3.6  Epsilon and identity relations   cep 4024
2.3.7  Partial and complete ordering   wpo 4031
2.3.8  Founded and set-like relations   wfrfor 4064
2.3.9  Ordinals   word 4099
2.4  IZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 4170
2.4.2  Ordinals (continued)   ordon 4212
2.5  IZF Set Theory - add the Axiom of Set Induction
2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4257
2.5.2  Introduce the Axiom of Set Induction   ax-setind 4262
2.5.3  Transfinite induction   tfi 4305
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-iinf 4311
2.6.2  The natural numbers (i.e. finite ordinals)   com 4313
2.6.3  Peano's postulates   peano1 4317
2.6.4  Finite induction (for finite ordinals)   find 4322
2.6.5  The Natural Numbers (continued)   nn0suc 4327
2.6.6  Relations   cxp 4343
2.6.7  Definite description binder (inverted iota)   cio 4865
2.6.8  Functions   wfun 4896
2.6.9  Restricted iota (description binder)   crio 5467
2.6.10  Operations   co 5512
2.6.11  "Maps to" notation   elmpt2cl 5698
2.6.12  Function operation   cof 5710
2.6.13  Functions (continued)   resfunexgALT 5737
2.6.14  First and second members of an ordered pair   c1st 5765
*2.6.15  Special "Maps to" operations   mpt2xopn0yelv 5854
2.6.16  Function transposition   ctpos 5859
2.6.17  Undefined values   pwuninel2 5897
2.6.18  Functions on ordinals; strictly monotone ordinal functions   iunon 5899
2.6.19  "Strong" transfinite recursion   crecs 5919
2.6.20  Recursive definition generator   crdg 5956
2.6.21  Finite recursion   cfrec 5977
2.6.22  Ordinal arithmetic   c1o 5994
2.6.23  Natural number arithmetic   nna0 6053
2.6.24  Equivalence relations and classes   wer 6103
2.6.25  Equinumerosity   cen 6219
2.6.26  Pigeonhole Principle   phplem1 6315
2.6.27  Finite sets   fidceq 6330
2.6.28  Ordinal isomorphism   ordiso2 6357
2.6.29  Cardinal numbers   ccrd 6359
*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 6370
3.1.2  Final derivation of real and complex number postulates   axcnex 6935
3.1.3  Real and complex number postulates restated as axioms   ax-cnex 6975
3.2  Derive the basic properties from the field axioms
3.2.1  Some deductions from the field axioms for complex numbers   cnex 7005
3.2.2  Infinity and the extended real number system   cpnf 7057
3.2.3  Restate the ordering postulates with extended real "less than"   axltirr 7086
3.2.4  Ordering on reals   lttr 7092
3.2.5  Initial properties of the complex numbers   mul12 7142
3.3  Real and complex numbers - basic operations
3.3.1  Addition   add12 7169
3.3.2  Subtraction   cmin 7182
3.3.3  Multiplication   kcnktkm1cn 7380
3.3.4  Ordering on reals (cont.)   ltadd2 7416
3.3.5  Real Apartness   creap 7565
3.3.6  Complex Apartness   cap 7572
3.3.7  Reciprocals   recextlem1 7632
3.3.8  Division   cdiv 7651
3.3.9  Ordering on reals (cont.)   ltp1 7810
3.3.10  Imaginary and complex number properties   crap0 7910
3.4  Integer sets
3.4.1  Positive integers (as a subset of complex numbers)   cn 7914
3.4.2  Principle of mathematical induction   nnind 7930
*3.4.3  Decimal representation of numbers   c2 7964
*3.4.4  Some properties of specific numbers   neg1cn 8022
3.4.5  Simple number properties   halfcl 8151
3.4.6  The Archimedean property   arch 8178
3.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 8181
3.4.8  Integers (as a subset of complex numbers)   cz 8245
3.4.9  Decimal arithmetic   cdc 8368
3.4.10  Upper sets of integers   cuz 8473
3.4.11  Rational numbers (as a subset of complex numbers)   cq 8554
3.4.12  Complex numbers as pairs of reals   cnref1o 8582
3.5  Order sets
3.5.1  Positive reals (as a subset of complex numbers)   crp 8583
3.5.2  Infinity and the extended real number system (cont.)   cxne 8686
3.5.3  Real number intervals   cioo 8757
3.5.4  Finite intervals of integers   cfz 8874
*3.5.5  Finite intervals of nonnegative integers   elfz2nn0 8973
3.5.6  Half-open integer ranges   cfzo 8999
3.5.7  Rational numbers (cont.)   qtri3or 9098
3.6  Elementary integer functions
3.6.1  The floor and ceiling functions   cfl 9112
3.6.2  The modulo (remainder) operation   cmo 9164
3.6.3  Miscellaneous theorems about integers   frec2uz0d 9185
3.6.4  The infinite sequence builder "seq"   cseq 9211
3.6.5  Integer powers   cexp 9254
3.7  Elementary real and complex functions
3.7.1  The "shift" operation   cshi 9415
3.7.2  Real and imaginary parts; conjugate   ccj 9439
3.7.3  Sequence convergence   caucvgrelemrec 9578
3.7.4  Square root; absolute value   csqrt 9594
3.8  Elementary limits and convergence
3.8.1  Limits   cli 9799
3.8.2  Finite and infinite sums   csu 9872
*PART 4  ELEMENTARY NUMBER THEORY
4.1  Elementary properties of divisibility
4.1.1  Rationality of square root of 2   sqr2irrlem 9877
4.1.2  Algorithms   nn0seqcvgd 9880
PART 5  GUIDES AND MISCELLANEA
5.1  Guides (conventions, explanations, and examples)
*5.1.1  Conventions   conventions 9892
5.1.2  Definitional examples   ex-or 9893
PART 6  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
6.1  Mathboxes for user contributions
6.1.1  Mathbox guidelines   mathbox 9897
6.2  Mathbox for Mykola Mostovenko
6.3  Mathbox for BJ
6.3.1  Propositional calculus   nnexmid 9899
6.3.2  Predicate calculus   bj-ex 9902
*6.3.3  Extensionality   bj-vtoclgft 9914
*6.3.4  Bounded formulas   wbd 9932
*6.3.5  Bounded classes   wbdc 9960
*6.3.6  Bounded separation   ax-bdsep 10004
6.3.6.1  Delta_0-classical logic   ax-bj-d0cl 10044
6.3.6.2  Inductive classes and the class of natural numbers (finite ordinals)   wind 10050
*6.3.6.3  The first three Peano postulates   bj-peano2 10063
*6.3.7  Axiom of infinity   ax-infvn 10066
*6.3.7.1  The set of natural numbers (finite ordinals)   ax-infvn 10066
*6.3.7.2  Peano's fifth postulate   bdpeano5 10068
*6.3.7.3  Bounded induction and Peano's fourth postulate   findset 10070
*6.3.8  Set induction   setindft 10090
*6.3.8.1  Set induction   setindft 10090
*6.3.8.2  Full induction   bj-findis 10104
*6.3.9  Strong collection   ax-strcoll 10107
*6.3.10  Subset collection   ax-sscoll 10112
6.3.11  Real numbers   ax-ddkcomp 10114
6.4  Mathbox for David A. Wheeler
*6.4.1  Allsome quantifier   walsi 10115
6.5  Mathbox for Jim Kingdon

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