HomeHome Intuitionistic Logic Explorer
Theorem List (Table of Contents)
< Wrap  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page:  Detailed Table of Contents  Page List

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 real and complex functions
PART 4  GUIDES AND MISCELLANEA
      4.1  Guides (conventions, explanations, and examples)
PART 5  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      5.1  Mathboxes for user contributions
      5.2  Mathbox for Mykola Mostovenko
      5.3  Mathbox for BJ
      5.4  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   dummylink 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 628
            1.2.7  Stable propositions   wstab 738
            1.2.8  Decidable propositions   wdc 741
            *1.2.9  Theorems of decidable propositions   condc 748
            1.2.10  Testable propositions   dftest 821
            1.2.11  Miscellaneous theorems of propositional calculus   pm5.21nd 824
            1.2.12  Abbreviated conjunction and disjunction of three wff's   w3o 883
            1.2.13  True and false constants   wal 1240
                  *1.2.13.1  Universal quantifier for use by df-tru   wal 1240
                  *1.2.13.2  Equality predicate for use by df-tru   cv 1241
                  1.2.13.3  Define the true and false constants   wtru 1243
            1.2.14  Logical 'xor'   wxo 1265
            *1.2.15  Truth tables: Operations on true and false constants   truantru 1289
            *1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1311
            1.2.17  Logical implication (continued)   syl6an 1320
      1.3  Predicate calculus mostly without distinct variables
            *1.3.1  Universal quantifier (continued)   ax-5 1333
            *1.3.2  Equality predicate (continued)   weq 1389
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1416
            1.3.4  Introduce Axiom of Existence   ax-i9 1420
            1.3.5  Additional intuitionistic axioms   ax-ial 1424
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1426
            1.3.7  The existential quantifier   19.8a 1479
            1.3.8  Equality theorems without distinct variables   a9e 1583
            1.3.9  Axioms ax-10 and ax-11   ax10o 1600
            1.3.10  Substitution (without distinct variables)   wsb 1642
            1.3.11  Theorems using axiom ax-11   equs5a 1672
      1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1689
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1700
            1.4.3  More theorems related to ax-11 and substitution   albidv 1702
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1735
            1.4.5  More substitution theorems   hbs1 1811
            1.4.6  Existential uniqueness   weu 1897
            *1.4.7  Aristotelian logic: Assertic syllogisms   barbara 1995
*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 2019
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2023
            2.1.3  Class form not-free predicate   wnfc 2162
            2.1.4  Negated equality and membership   wne 2201
                  2.1.4.1  Negated equality   neii 2205
                  2.1.4.2  Negated membership   neli 2293
            2.1.5  Restricted quantification   wral 2300
            2.1.6  The universal class   cvv 2551
            *2.1.7  Conditional equality (experimental)   wcdeq 2741
            2.1.8  Russell's Paradox   ru 2757
            2.1.9  Proper substitution of classes for sets   wsbc 2758
            2.1.10  Proper substitution of classes for sets into classes   csb 2846
            2.1.11  Define basic set operations and relations   cdif 2908
            2.1.12  Subclasses and subsets   df-ss 2925
            2.1.13  The difference, union, and intersection of two classes   difeq1 3049
                  2.1.13.1  The difference of two classes   difeq1 3049
                  2.1.13.2  The union of two classes   elun 3078
                  2.1.13.3  The intersection of two classes   elin 3120
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3161
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3197
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3211
            2.1.14  The empty set   c0 3218
            2.1.15  Conditional operator   cif 3325
            2.1.16  Power classes   cpw 3351
            2.1.17  Unordered and ordered pairs   csn 3367
            2.1.18  The union of a class   cuni 3571
            2.1.19  The intersection of a class   cint 3606
            2.1.20  Indexed union and intersection   ciun 3648
            2.1.21  Disjointness   wdisj 3736
            2.1.22  Binary relations   wbr 3755
            2.1.23  Ordered-pair class abstractions (class builders)   copab 3808
            2.1.24  Transitive classes   wtr 3845
      2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 3863
            2.2.2  Introduce the Axiom of Separation   ax-sep 3866
            2.2.3  Derive the Null Set Axiom   zfnuleu 3872
            2.2.4  Theorems requiring subset and intersection existence   nalset 3878
            2.2.5  Theorems requiring empty set existence   class2seteq 3907
            2.2.6  Collection principle   bnd 3916
      2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 3918
            2.3.2  Axiom of Pairing   ax-pr 3935
            2.3.3  Ordered pair theorem   opm 3962
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 3985
            2.3.5  Power class of union and intersection   pwin 4010
            2.3.6  Epsilon and identity relations   cep 4015
            2.3.7  Partial and complete ordering   wpo 4022
            2.3.8  Set-like relations   wse 4055
            2.3.9  Ordinals   word 4065
      2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4136
            2.4.2  Ordinals (continued)   ordon 4178
      2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4217
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4220
            2.5.3  Transfinite induction   tfi 4248
      2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4254
            2.6.2  The natural numbers (i.e. finite ordinals)   com 4256
            2.6.3  Peano's postulates   peano1 4260
            2.6.4  Finite induction (for finite ordinals)   find 4265
            2.6.5  The Natural Numbers (continued)   nn0suc 4270
            2.6.6  Relations   cxp 4286
            2.6.7  Definite description binder (inverted iota)   cio 4808
            2.6.8  Functions   wfun 4839
            2.6.9  Restricted iota (description binder)   crio 5410
            2.6.10  Operations   co 5455
            2.6.11  "Maps to" notation   elmpt2cl 5640
            2.6.12  Function operation   cof 5652
            2.6.13  Functions (continued)   resfunexgALT 5679
            2.6.14  First and second members of an ordered pair   c1st 5707
            *2.6.15  Special "Maps to" operations   mpt2xopn0yelv 5795
            2.6.16  Function transposition   ctpos 5800
            2.6.17  Undefined values   pwuninel2 5838
            2.6.18  Functions on ordinals; strictly monotone ordinal functions   iunon 5840
            2.6.19  "Strong" transfinite recursion   crecs 5860
            2.6.20  Recursive definition generator   crdg 5896
            2.6.21  Finite recursion   cfrec 5917
            2.6.22  Ordinal arithmetic   c1o 5933
            2.6.23  Natural number arithmetic   nna0 5992
            2.6.24  Equivalence relations and classes   wer 6039
            2.6.25  Equinumerosity   cen 6155
            2.6.26  Finite sets   nnfi 6251
*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 6256
            3.1.2  Final derivation of real and complex number postulates   axcnex 6725
            3.1.3  Real and complex number postulates restated as axioms   ax-cnex 6754
      3.2  Derive the basic properties from the field axioms
            3.2.1  Some deductions from the field axioms for complex numbers   cnex 6783
            3.2.2  Infinity and the extended real number system   cpnf 6834
            3.2.3  Restate the ordering postulates with extended real "less than"   axltirr 6863
            3.2.4  Ordering on reals   lttr 6869
            3.2.5  Initial properties of the complex numbers   mul12 6919
      3.3  Real and complex numbers - basic operations
            3.3.1  Addition   add12 6946
            3.3.2  Subtraction   cmin 6959
            3.3.3  Multiplication   kcnktkm1cn 7156
            3.3.4  Ordering on reals (cont.)   ltadd2 7192
            3.3.5  Real Apartness   creap 7338
            3.3.6  Complex Apartness   cap 7345
            3.3.7  Reciprocals   recextlem1 7394
            3.3.8  Division   cdiv 7413
            3.3.9  Ordering on reals (cont.)   ltp1 7571
            3.3.10  Imaginary and complex number properties   crap0 7671
      3.4  Integer sets
            3.4.1  Positive integers (as a subset of complex numbers)   cn 7675
            3.4.2  Principle of mathematical induction   nnind 7691
            *3.4.3  Decimal representation of numbers   c2 7724
            *3.4.4  Some properties of specific numbers   neg1cn 7780
            3.4.5  Simple number properties   halfcl 7908
            3.4.6  The Archimedean property   arch 7934
            3.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 7937
            3.4.8  Integers (as a subset of complex numbers)   cz 8001
            3.4.9  Decimal arithmetic   cdc 8124
            3.4.10  Upper sets of integers   cuz 8229
            3.4.11  Rational numbers (as a subset of complex numbers)   cq 8310
            3.4.12  Complex numbers as pairs of reals   cnref1o 8337
      3.5  Order sets
            3.5.1  Positive reals (as a subset of complex numbers)   crp 8338
            3.5.2  Infinity and the extended real number system (cont.)   cxne 8436
            3.5.3  Real number intervals   cioo 8507
            3.5.4  Finite intervals of integers   cfz 8624
            *3.5.5  Finite intervals of nonnegative integers   elfz2nn0 8723
            3.5.6  Half-open integer ranges   cfzo 8749
            3.5.7  Miscellaneous theorems about integers   frec2uz0d 8846
            3.5.8  The infinite sequence builder "seq"   cseq 8872
            3.5.9  Integer powers   cexp 8888
      3.6  Elementary real and complex functions
            3.6.1  Real and imaginary parts; conjugate   ccj 9047
            3.6.2  Square root; absolute value   csqrt 9185
PART 4  GUIDES AND MISCELLANEA
      4.1  Guides (conventions, explanations, and examples)
            *4.1.1  Conventions   conventions 9211
PART 5  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      5.1  Mathboxes for user contributions
            5.1.1  Mathbox guidelines   mathbox 9212
      5.2  Mathbox for Mykola Mostovenko
      5.3  Mathbox for BJ
            5.3.1  Propositional calculus   nnexmid 9214
            5.3.2  Predicate calculus   bj-ex 9217
            *5.3.3  Extensionality   bj-vtoclgft 9229
            *5.3.4  Bounded formulas   wbd 9247
            *5.3.5  Bounded classes   wbdc 9275
            *5.3.6  Bounded separation   ax-bdsep 9319
                  5.3.6.1  Delta_0-classical logic   ax-bj-d0cl 9355
                  5.3.6.2  Inductive classes and the class of natural numbers (finite ordinals)   wind 9361
                  *5.3.6.3  The first three Peano postulates   bj-peano2 9373
            *5.3.7  Axiom of infinity   ax-infvn 9375
                  *5.3.7.1  The set of natural numbers (finite ordinals)   ax-infvn 9375
                  *5.3.7.2  The remaining two Peano postulates   bdpeano5 9377
                  *5.3.7.3  Bounded induction   findset 9379
            *5.3.8  Set induction   setindft 9395
                  *5.3.8.1  Set induction   setindft 9395
                  *5.3.8.2  Full induction   bj-findis 9409
            *5.3.9  Strong collection   ax-strcoll 9412
            *5.3.10  Subset collection   ax-sscoll 9417
      5.4  Mathbox for David A. Wheeler
            *5.4.1  Allsome quantifier   walsi 9419

    < Wrap  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9427
  Copyright terms: Public domain < Wrap  Next >