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
      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 604
            1.2.6  Logical disjunction   wo 698
            1.2.7  Stable propositions   wstab 816
            1.2.8  Decidable propositions   wdc 820
            ​*1.2.9  Theorems of decidable propositions   const 838
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 902
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 962
            1.2.12  True and false constants   wal 1330
                  ​*1.2.12.1  Universal quantifier for use by df-tru   wal 1330
                  ​*1.2.12.2  Equality predicate for use by df-tru   cv 1331
                  1.2.12.3  Define the true and false constants   wtru 1333
            1.2.13  Logical 'xor'   wxo 1354
            ​*1.2.14  Truth tables: Operations on true and false constants   truantru 1380
            ​*1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1402
            1.2.16  Logical implication (continued)   syl6an 1411
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1424
            ​*1.3.2  Equality predicate (continued)   weq 1480
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1503
            1.3.4  Introduce Axiom of Existence   ax-i9 1507
            1.3.5  Additional intuitionistic axioms   ax-ial 1511
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1513
            1.3.7  The existential quantifier   19.8a 1567
            1.3.8  Equality theorems without distinct variables   a9e 1673
            1.3.9  Axioms ax-10 and ax-11   ax10o 1692
            1.3.10  Substitution (without distinct variables)   wsb 1739
            1.3.11  Theorems using axiom ax-11   equs5a 1771
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1788
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1799
            1.4.3  More theorems related to ax-11 and substitution   albidv 1801
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1835
            1.4.5  More substitution theorems   hbs1 1915
            1.4.6  Existential uniqueness   weu 2003
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2101
      ​​*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 2136
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2140
            2.1.3  Class form not-free predicate   wnfc 2283
            2.1.4  Negated equality and membership   wne 2324
                  2.1.4.1  Negated equality   wne 2324
                  2.1.4.2  Negated membership   wnel 2419
            2.1.5  Restricted quantification   wral 2432
            2.1.6  The universal class   cvv 2709
            ​*2.1.7  Conditional equality (experimental)   wcdeq 2916
            2.1.8  Russell's Paradox   ru 2932
            2.1.9  Proper substitution of classes for sets   wsbc 2933
            2.1.10  Proper substitution of classes for sets into classes   csb 3027
            2.1.11  Define basic set operations and relations   cdif 3095
            2.1.12  Subclasses and subsets   df-ss 3111
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3213
                  2.1.13.1  The difference of two classes   dfdif3 3213
                  2.1.13.2  The union of two classes   elun 3244
                  2.1.13.3  The intersection of two classes   elin 3286
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3334
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3369
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3383
            2.1.14  The empty set   c0 3390
            2.1.15  Conditional operator   cif 3501
            2.1.16  Power classes   cpw 3539
            2.1.17  Unordered and ordered pairs   csn 3556
            2.1.18  The union of a class   cuni 3768
            2.1.19  The intersection of a class   cint 3803
            2.1.20  Indexed union and intersection   ciun 3845
            2.1.21  Disjointness   wdisj 3938
            2.1.22  Binary relations   wbr 3961
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4020
            2.1.24  Transitive classes   wtr 4058
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4075
            2.2.2  Introduce the Axiom of Separation   ax-sep 4078
            2.2.3  Derive the Null Set Axiom   zfnuleu 4084
            2.2.4  Theorems requiring subset and intersection existence   nalset 4090
            2.2.5  Theorems requiring empty set existence   class2seteq 4119
            2.2.6  Collection principle   bnd 4128
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4130
            2.3.2  A notation for excluded middle   wem 4150
            2.3.3  Axiom of Pairing   ax-pr 4164
            2.3.4  Ordered pair theorem   opm 4189
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4212
            2.3.6  Power class of union and intersection   pwin 4237
            2.3.7  Epsilon and identity relations   cep 4242
            ​*2.3.8  Partial and total orderings   wpo 4249
            2.3.9  Founded and set-like relations   wfrfor 4282
            2.3.10  Ordinals   word 4317
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4388
            2.4.2  Ordinals (continued)   ordon 4439
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4485
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4490
            2.5.3  Transfinite induction   tfi 4535
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4541
            2.6.2  The natural numbers   com 4543
            2.6.3  Peano's postulates   peano1 4547
            2.6.4  Finite induction (for finite ordinals)   find 4552
            2.6.5  The Natural Numbers (continued)   nn0suc 4557
            2.6.6  Relations   cxp 4577
            2.6.7  Definite description binder (inverted iota)   cio 5126
            2.6.8  Functions   wfun 5157
            2.6.9  Restricted iota (description binder)   crio 5769
            2.6.10  Operations   co 5814
            2.6.11  Maps-to notation   elmpocl 6008
            2.6.12  Function operation   cof 6020
            2.6.13  Functions (continued)   resfunexgALT 6048
            2.6.14  First and second members of an ordered pair   c1st 6076
            ​*2.6.15  Special maps-to operations   opeliunxp2f 6175
            2.6.16  Function transposition   ctpos 6181
            2.6.17  Undefined values   pwuninel2 6219
            2.6.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6221
            2.6.19  "Strong" transfinite recursion   crecs 6241
            2.6.20  Recursive definition generator   crdg 6306
            2.6.21  Finite recursion   cfrec 6327
            2.6.22  Ordinal arithmetic   c1o 6346
            2.6.23  Natural number arithmetic   nna0 6410
            2.6.24  Equivalence relations and classes   wer 6466
            2.6.25  The mapping operation   cmap 6582
            2.6.26  Infinite Cartesian products   cixp 6632
            2.6.27  Equinumerosity   cen 6672
            2.6.28  Equinumerosity (cont.)   xpf1o 6778
            2.6.29  Pigeonhole Principle   phplem1 6786
            2.6.30  Finite sets   fict 6802
            2.6.31  Schroeder-Bernstein Theorem   sbthlem1 6890
            2.6.32  Finite intersections   cfi 6901
            2.6.33  Supremum and infimum   csup 6914
            2.6.34  Ordinal isomorphism   ordiso2 6965
            2.6.35  Disjoint union   cdju 6967
                  2.6.35.1  Disjoint union   cdju 6967
                  ​*2.6.35.2  Left and right injections of a disjoint union   cinl 6975
                  2.6.35.3  Universal property of the disjoint union   djuss 7000
                  2.6.35.4  Dominance and equinumerosity properties of disjoint union   djudom 7023
                  2.6.35.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7032
                  2.6.35.6  Countable sets   0ct 7037
            ​*2.6.36  The one-point compactification of the natural numbers   xnninf 7049
            2.6.37  Omniscient sets   comni 7056
            2.6.38  Markov's principle   cmarkov 7073
            2.6.39  Weakly omniscient sets   cwomni 7085
            2.6.40  Cardinal numbers   ccrd 7093
            2.6.41  Axiom of Choice equivalents   wac 7119
            2.6.42  Cardinal number arithmetic   endjudisj 7124
            2.6.43  Ordinal trichotomy   exmidontriimlem1 7135
            2.6.44  Excluded middle and the power set of a singleton   pw1on 7140
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7161
​​*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 7171
            4.1.2  Final derivation of real and complex number postulates   axcnex 7758
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7802
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 7835
            4.2.2  Infinity and the extended real number system   cpnf 7888
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 7923
            4.2.4  Ordering on reals   lttr 7930
            4.2.5  Initial properties of the complex numbers   mul12 7983
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8012
            4.3.2  Subtraction   cmin 8025
            4.3.3  Multiplication   kcnktkm1cn 8237
            4.3.4  Ordering on reals (cont.)   ltadd2 8273
            4.3.5  Real Apartness   creap 8428
            4.3.6  Complex Apartness   cap 8435
            4.3.7  Reciprocals   recextlem1 8504
            4.3.8  Division   cdiv 8524
            4.3.9  Ordering on reals (cont.)   ltp1 8694
            4.3.10  Suprema   lbreu 8795
            4.3.11  Imaginary and complex number properties   crap0 8808
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 8812
            4.4.2  Principle of mathematical induction   nnind 8828
            ​*4.4.3  Decimal representation of numbers   c2 8863
            ​*4.4.4  Some properties of specific numbers   neg1cn 8917
            4.4.5  Simple number properties   halfcl 9038
            4.4.6  The Archimedean property   arch 9066
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9069
            ​*4.4.8  Extended nonnegative integers   cxnn0 9132
            4.4.9  Integers (as a subset of complex numbers)   cz 9146
            4.4.10  Decimal arithmetic   cdc 9274
            4.4.11  Upper sets of integers   cuz 9418
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9506
            4.4.13  Complex numbers as pairs of reals   cnref1o 9537
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9538
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9654
            4.5.3  Real number intervals   cioo 9770
            4.5.4  Finite intervals of integers   cfz 9890
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 9992
            4.5.6  Half-open integer ranges   cfzo 10019
            4.5.7  Rational numbers (cont.)   qtri3or 10120
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10145
            4.6.2  The modulo (remainder) operation   cmo 10199
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10276
            4.6.4  Strong induction over upper sets of integers   uzsinds 10319
            4.6.5  The infinite sequence builder "seq"   cseq 10322
            4.6.6  Integer powers   cexp 10396
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10570
            4.6.8  Factorial function   cfa 10576
            4.6.9  The binomial coefficient operation   cbc 10598
            4.6.10  The ` # ` (set size) function   chash 10626
      ​4.7  Elementary real and complex functions
            4.7.1  The "shift" operation   cshi 10691
            4.7.2  Real and imaginary parts; conjugate   ccj 10716
            4.7.3  Sequence convergence   caucvgrelemrec 10856
            4.7.4  Square root; absolute value   csqrt 10873
            4.7.5  The maximum of two real numbers   maxcom 11080
            4.7.6  The minimum of two real numbers   mincom 11105
            4.7.7  The maximum of two extended reals   xrmaxleim 11118
            4.7.8  The minimum of two extended reals   xrnegiso 11136
      ​4.8  Elementary limits and convergence
            4.8.1  Limits   cli 11152
            4.8.2  Finite and infinite sums   csu 11227
            4.8.3  The binomial theorem   binomlem 11357
            4.8.4  Infinite sums (cont.)   isumshft 11364
            4.8.5  Miscellaneous converging and diverging sequences   divcnv 11371
            4.8.6  Arithmetic series   arisum 11372
            4.8.7  Geometric series   expcnvap0 11376
            4.8.8  Ratio test for infinite series convergence   cvgratnnlembern 11397
            4.8.9  Mertens' theorem   mertenslemub 11408
            4.8.10  Finite and infinite products   prodf 11412
                  4.8.10.1  Product sequences   prodf 11412
                  4.8.10.2  Non-trivial convergence   ntrivcvgap 11422
                  4.8.10.3  Complex products   cprod 11424
                  4.8.10.4  Finite products   fprodseq 11457
      ​4.9  Elementary trigonometry
            4.9.1  The exponential, sine, and cosine functions   ce 11516
                  4.9.1.1  The circle constant (tau = 2 pi)   ctau 11648
            4.9.2  _e is irrational   eirraplem 11650
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11660
            ​*5.1.2  Even and odd numbers   evenelz 11731
            5.1.3  The division algorithm   divalglemnn 11782
            5.1.4  The greatest common divisor operator   cgcd 11802
            5.1.5  Bézout's identity   bezoutlemnewy 11852
            5.1.6  Algorithms   nn0seqcvgd 11890
            5.1.7  Euclid's Algorithm   eucalgval2 11902
            ​*5.1.8  The least common multiple   clcm 11909
            ​*5.1.9  Coprimality and Euclid's lemma   coprmgcdb 11937
            5.1.10  Cancellability of congruences   congr 11949
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 11956
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 11990
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12007
            5.2.4  Properties of the canonical representation of a rational   cnumer 12027
            5.2.5  Euler's theorem   cphi 12054
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12080
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12133
            6.1.2  Slot definitions   cplusg 12199
            6.1.3  Definition of the structure product   crest 12298
      ​6.2  The complex numbers as an algebraic extensible structure
            6.2.1  Definition and basic properties   cpsmet 12326
​PART 7  BASIC TOPOLOGY
      ​7.1  Topology
            ​*7.1.1  Topological spaces   ctop 12342
                  7.1.1.1  Topologies   ctop 12342
                  7.1.1.2  Topologies on sets   ctopon 12355
                  7.1.1.3  Topological spaces   ctps 12375
            7.1.2  Topological bases   ctb 12387
            7.1.3  Examples of topologies   distop 12432
            7.1.4  Closure and interior   ccld 12439
            7.1.5  Neighborhoods   cnei 12485
            7.1.6  Subspace topologies   restrcl 12514
            7.1.7  Limits and continuity in topological spaces   ccn 12532
            7.1.8  Product topologies   ctx 12599
            7.1.9  Continuous function-builders   cnmptid 12628
            7.1.10  Homeomorphisms   chmeo 12647
      ​7.2  Metric spaces
            7.2.1  Pseudometric spaces   psmetrel 12669
            7.2.2  Basic metric space properties   cxms 12683
            7.2.3  Metric space balls   blfvalps 12732
            7.2.4  Open sets of a metric space   mopnrel 12788
            7.2.5  Continuity in metric spaces   metcnp3 12858
            7.2.6  Topology on the reals   qtopbasss 12868
            7.2.7  Topological definitions using the reals   ccncf 12904
​PART 8  BASIC REAL AND COMPLEX ANALYSIS
            8.0.1  Dedekind cuts   dedekindeulemuub 12942
            8.0.2  Intermediate value theorem   ivthinclemlm 12959
      ​8.1  Derivatives
            8.1.1  Real and complex differentiation   climc 12970
                  8.1.1.1  Derivatives of functions of one complex or real variable   climc 12970
​PART 9  BASIC REAL AND COMPLEX FUNCTIONS
      ​9.1  Basic trigonometry
            9.1.1  The exponential, sine, and cosine functions (cont.)   efcn 13036
            9.1.2  Properties of pi = 3.14159...   pilem1 13047
            9.1.3  The natural logarithm on complex numbers   clog 13124
            ​*9.1.4  Logarithms to an arbitrary base   clogb 13207
​PART 10  GUIDES AND MISCELLANEA
      ​10.1  Guides (conventions, explanations, and examples)
            ​*10.1.1  Conventions   conventions 13243
            10.1.2  Definitional examples   ex-or 13244
​PART 11  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​11.1  Mathboxes for user contributions
            11.1.1  Mathbox guidelines   mathbox 13254
      ​11.2  Mathbox for BJ
            11.2.1  Propositional calculus   bj-nnsn 13255
                  ​*11.2.1.1  Stable formulas   bj-trst 13260
                  11.2.1.2  Decidable formulas   bj-trdc 13270
            11.2.2  Predicate calculus   bj-ex 13282
            11.2.3  Set theorey miscellaneous   bj-el2oss1o 13294
            ​*11.2.4  Extensionality   bj-vtoclgft 13295
            ​*11.2.5  Decidability of classes   wdcin 13313
            11.2.6  Disjoint union   djucllem 13320
            11.2.7  Miscellaneous   2ssom 13323
            ​*11.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 13333
                  *11.2.8.1  Bounded formulas   wbd 13333
                  ​*11.2.8.2  Bounded classes   wbdc 13361
            ​*11.2.9  CZF: Bounded separation   ax-bdsep 13405
                  11.2.9.1  Delta_0-classical logic   ax-bj-d0cl 13445
                  11.2.9.2  Inductive classes and the class of natural number ordinals   wind 13447
                  ​*11.2.9.3  The first three Peano postulates   bj-peano2 13460
            ​*11.2.10  CZF: Infinity   ax-infvn 13462
                  *11.2.10.1  The set of natural number ordinals   ax-infvn 13462
                  ​*11.2.10.2  Peano's fifth postulate   bdpeano5 13464
                  ​*11.2.10.3  Bounded induction and Peano's fourth postulate   findset 13466
            ​*11.2.11  CZF: Set induction   setindft 13486
                  *11.2.11.1  Set induction   setindft 13486
                  ​*11.2.11.2  Full induction   bj-findis 13500
            ​*11.2.12  CZF: Strong collection   ax-strcoll 13503
            ​*11.2.13  CZF: Subset collection   ax-sscoll 13508
            11.2.14  Real numbers   ax-ddkcomp 13510
      ​11.3  Mathbox for Jim Kingdon
            11.3.1  Natural numbers   el2oss1o 13511
            11.3.2  The power set of a singleton   pwtrufal 13516
            11.3.3  Omniscience of NN+oo   0nninf 13523
            11.3.4  Schroeder-Bernstein Theorem   exmidsbthrlem 13542
            11.3.5  Real and complex numbers   qdencn 13547
            ​*11.3.6  Analytic omniscience principles   trilpolemclim 13556
            11.3.7  Supremum and infimum   supfz 13588
            11.3.8  Circle constant   taupi 13590
      ​11.4  Mathbox for Mykola Mostovenko
      ​11.5  Mathbox for David A. Wheeler
            11.5.1  Testable propositions   dftest 13592
            ​*11.5.2  Allsome quantifier   walsi 13593