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
2.2  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

(* 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
2.1.1  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 2610
*2.1.7  Conditional equality (experimental)   wcdeq 2808
2.1.9  Proper substitution of classes for sets   wsbc 2825
2.1.10  Proper substitution of classes for sets into classes   csb 2918
2.1.11  Define basic set operations and relations   cdif 2980
2.1.12  Subclasses and subsets   df-ss 2996
2.1.13  The difference, union, and intersection of two classes   dfdif3 3093
2.1.13.1  The difference of two classes   dfdif3 3093
2.1.13.2  The union of two classes   elun 3124
2.1.13.3  The intersection of two classes   elin 3166
2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3213
2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3247
2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3261
2.1.14  The empty set   c0 3268
2.1.15  Conditional operator   cif 3369
2.1.16  Power classes   cpw 3401
2.1.17  Unordered and ordered pairs   csn 3417
2.1.18  The union of a class   cuni 3622
2.1.19  The intersection of a class   cint 3657
2.1.20  Indexed union and intersection   ciun 3699
2.1.21  Disjointness   wdisj 3787
2.1.22  Binary relations   wbr 3806
2.1.23  Ordered-pair class abstractions (class builders)   copab 3859
2.1.24  Transitive classes   wtr 3896
2.2  IZF Set Theory - add the Axioms of Collection and Separation
2.2.1  Introduce the Axiom of Collection   ax-coll 3914
2.2.2  Introduce the Axiom of Separation   ax-sep 3917
2.2.3  Derive the Null Set Axiom   zfnuleu 3923
2.2.4  Theorems requiring subset and intersection existence   nalset 3929
2.2.5  Theorems requiring empty set existence   class2seteq 3958
2.2.6  Collection principle   bnd 3967
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
2.3.1  Introduce the Axiom of Power Sets   ax-pow 3969
2.3.2  A notation for excluded middle   wem 3986
2.3.3  Axiom of Pairing   ax-pr 3993
2.3.4  Ordered pair theorem   opm 4018
2.3.5  Ordered-pair class abstractions (cont.)   opabid 4041
2.3.6  Power class of union and intersection   pwin 4066
2.3.7  Epsilon and identity relations   cep 4071
2.3.8  Partial and complete ordering   wpo 4078
2.3.9  Founded and set-like relations   wfrfor 4111
2.3.10  Ordinals   word 4146
2.4  IZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 4217
2.4.2  Ordinals (continued)   ordon 4259
2.5  IZF Set Theory - add the Axiom of Set Induction
2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4304
2.5.2  Introduce the Axiom of Set Induction   ax-setind 4309
2.5.3  Transfinite induction   tfi 4352
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-iinf 4358
2.6.2  The natural numbers (i.e. finite ordinals)   com 4360
2.6.3  Peano's postulates   peano1 4364
2.6.4  Finite induction (for finite ordinals)   find 4369
2.6.5  The Natural Numbers (continued)   nn0suc 4374
2.6.6  Relations   cxp 4390
2.6.7  Definite description binder (inverted iota)   cio 4916
2.6.8  Functions   wfun 4947
2.6.9  Restricted iota (description binder)   crio 5519
2.6.10  Operations   co 5564
2.6.11  "Maps to" notation   elmpt2cl 5750
2.6.12  Function operation   cof 5762
2.6.13  Functions (continued)   resfunexgALT 5789
2.6.14  First and second members of an ordered pair   c1st 5817
*2.6.15  Special "Maps to" operations   mpt2xopn0yelv 5909
2.6.16  Function transposition   ctpos 5914
2.6.17  Undefined values   pwuninel2 5952
2.6.18  Functions on ordinals; strictly monotone ordinal functions   iunon 5954
2.6.19  "Strong" transfinite recursion   crecs 5974
2.6.20  Recursive definition generator   crdg 6039
2.6.21  Finite recursion   cfrec 6060
2.6.22  Ordinal arithmetic   c1o 6079
2.6.23  Natural number arithmetic   nna0 6139
2.6.24  Equivalence relations and classes   wer 6191
2.6.25  Equinumerosity   cen 6307
2.6.26  Equinumerosity (cont.)   xpf1o 6407
2.6.27  Pigeonhole Principle   phplem1 6409
2.6.28  Finite sets   fict 6425
2.6.29  Supremum and infimum   csup 6490
2.6.30  Ordinal isomorphism   ordiso2 6541
2.6.31  Cardinal numbers   ccrd 6543
*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 6560
3.1.2  Final derivation of real and complex number postulates   axcnex 7125
3.1.3  Real and complex number postulates restated as axioms   ax-cnex 7165
3.2  Derive the basic properties from the field axioms
3.2.1  Some deductions from the field axioms for complex numbers   cnex 7195
3.2.2  Infinity and the extended real number system   cpnf 7248
3.2.3  Restate the ordering postulates with extended real "less than"   axltirr 7282
3.2.4  Ordering on reals   lttr 7288
3.2.5  Initial properties of the complex numbers   mul12 7340
3.3  Real and complex numbers - basic operations
3.3.2  Subtraction   cmin 7382
3.3.3  Multiplication   kcnktkm1cn 7590
3.3.4  Ordering on reals (cont.)   ltadd2 7626
3.3.5  Real Apartness   creap 7777
3.3.6  Complex Apartness   cap 7784
3.3.7  Reciprocals   recextlem1 7844
3.3.8  Division   cdiv 7863
3.3.9  Ordering on reals (cont.)   ltp1 8025
3.3.10  Suprema   lbreu 8126
3.3.11  Imaginary and complex number properties   crap0 8138
3.4  Integer sets
3.4.1  Positive integers (as a subset of complex numbers)   cn 8142
3.4.2  Principle of mathematical induction   nnind 8158
*3.4.3  Decimal representation of numbers   c2 8192
*3.4.4  Some properties of specific numbers   neg1cn 8247
3.4.5  Simple number properties   halfcl 8360
3.4.6  The Archimedean property   arch 8388
3.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 8391
*3.4.8  Extended nonnegative integers   cxnn0 8454
3.4.9  Integers (as a subset of complex numbers)   cz 8468
3.4.10  Decimal arithmetic   cdc 8594
3.4.11  Upper sets of integers   cuz 8736
3.4.12  Rational numbers (as a subset of complex numbers)   cq 8821
3.4.13  Complex numbers as pairs of reals   cnref1o 8850
3.5  Order sets
3.5.1  Positive reals (as a subset of complex numbers)   crp 8851
3.5.2  Infinity and the extended real number system (cont.)   cxne 8957
3.5.3  Real number intervals   cioo 9023
3.5.4  Finite intervals of integers   cfz 9141
*3.5.5  Finite intervals of nonnegative integers   elfz2nn0 9241
3.5.6  Half-open integer ranges   cfzo 9265
3.5.7  Rational numbers (cont.)   qtri3or 9365
3.6  Elementary integer functions
3.6.1  The floor and ceiling functions   cfl 9386
3.6.2  The modulo (remainder) operation   cmo 9440
3.6.3  Miscellaneous theorems about integers   frec2uz0d 9517
3.6.4  Strong induction over upper sets of integers   uzsinds 9554
3.6.5  The infinite sequence builder "seq"   cseq 9557
3.6.6  Integer powers   cexp 9608
3.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 9779
3.6.8  Factorial function   cfa 9785
3.6.9  The binomial coefficient operation   cbc 9807
3.6.10  The ` # ` (set size) function   chash 9835
3.7  Elementary real and complex functions
3.7.1  The "shift" operation   cshi 9887
3.7.2  Real and imaginary parts; conjugate   ccj 9911
3.7.3  Sequence convergence   caucvgrelemrec 10050
3.7.4  Square root; absolute value   csqrt 10067
3.7.5  The maximum of two real numbers   maxcom 10274
3.7.6  The minimum of two real numbers   mincom 10296
3.8  Elementary limits and convergence
3.8.1  Limits   cli 10302
3.8.2  Finite and infinite sums   csu 10375
*PART 4  ELEMENTARY NUMBER THEORY
4.1  Elementary properties of divisibility
4.1.1  The divides relation   cdvds 10387
*4.1.2  Even and odd numbers   evenelz 10458
4.1.3  The division algorithm   divalglemnn 10509
4.1.4  The greatest common divisor operator   cgcd 10529
4.1.5  Bézout's identity   bezoutlemnewy 10576
4.1.6  Algorithms   nn0seqcvgd 10614
4.1.7  Euclid's Algorithm   eucalgval2 10626
*4.1.8  The least common multiple   clcm 10633
*4.1.9  Coprimality and Euclid's lemma   coprmgcdb 10661
4.1.10  Cancellability of congruences   congr 10673
4.2  Elementary prime number theory
*4.2.1  Elementary properties   cprime 10680
*4.2.2  Coprimality and Euclid's lemma (cont.)   coprm 10714
4.2.3  Non-rationality of square root of 2   sqrt2irrlem 10731
4.2.4  Properties of the canonical representation of a rational   cnumer 10750
4.2.5  Euler's theorem   cphi 10777
4.3  Cardinality of real and complex number subsets
4.3.1  Countability of integers and rationals   oddennn 10796
PART 5  GUIDES AND MISCELLANEA
5.1  Guides (conventions, explanations, and examples)
*5.1.1  Conventions   conventions 10803
5.1.2  Definitional examples   ex-or 10804
PART 6  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
6.1  Mathboxes for user contributions
6.1.1  Mathbox guidelines   mathbox 10813
6.2  Mathbox for BJ
6.2.1  Propositional calculus   nnexmid 10814
6.2.2  Predicate calculus   bj-ex 10817
*6.2.3  Extensionality   bj-vtoclgft 10829
*6.2.4  Dedidability of classes   wdcin 10847
*6.2.5  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 10854
*6.2.5.1  Bounded formulas   wbd 10854
*6.2.5.2  Bounded classes   wbdc 10882
*6.2.6  CZF: Bounded separation   ax-bdsep 10926
6.2.6.1  Delta_0-classical logic   ax-bj-d0cl 10966
6.2.6.2  Inductive classes and the class of natural numbers (finite ordinals)   wind 10972
*6.2.6.3  The first three Peano postulates   bj-peano2 10985
*6.2.7  CZF: Infinity   ax-infvn 10987
*6.2.7.1  The set of natural numbers (finite ordinals)   ax-infvn 10987
*6.2.7.2  Peano's fifth postulate   bdpeano5 10989
*6.2.7.3  Bounded induction and Peano's fourth postulate   findset 10991
*6.2.8  CZF: Set induction   setindft 11011
*6.2.8.1  Set induction   setindft 11011
*6.2.8.2  Full induction   bj-findis 11025
*6.2.9  CZF: Strong collection   ax-strcoll 11028
*6.2.10  CZF: Subset collection   ax-sscoll 11033
6.2.11  Real numbers   ax-ddkcomp 11035
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 11038

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