Home Intuitionistic Logic ExplorerTheorem List (p. 131 of 133) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Axiomax-bdim 13001 An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED        BOUNDED        BOUNDED

Axiomax-bdan 13002 The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED        BOUNDED        BOUNDED

Axiomax-bdor 13003 The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED        BOUNDED        BOUNDED

Axiomax-bdn 13004 The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED        BOUNDED

Axiomax-bdal 13005* A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on . (Contributed by BJ, 25-Sep-2019.)
BOUNDED        BOUNDED

Axiomax-bdex 13006* A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on . (Contributed by BJ, 25-Sep-2019.)
BOUNDED        BOUNDED

Axiomax-bdeq 13007 An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
BOUNDED

Axiomax-bdel 13008 An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
BOUNDED

Axiomax-bdsb 13009 A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1736, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED

Theorembdeq 13010 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
BOUNDED BOUNDED

Theorembd0 13011 A formula equivalent to a bounded one is bounded. See also bd0r 13012. (Contributed by BJ, 3-Oct-2019.)
BOUNDED               BOUNDED

Theorembd0r 13012 A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13011) biconditional in the hypothesis, to work better with definitions ( is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
BOUNDED               BOUNDED

Theorembdbi 13013 A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED

Theorembdstab 13014 Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED STAB

Theorembddc 13015 Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED DECID

Theorembd3or 13016 A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED        BOUNDED

Theorembd3an 13017 A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED        BOUNDED

Theorembdth 13018 A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
BOUNDED

Theorembdtru 13019 The truth value is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED

Theorembdfal 13020 The truth value is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED

Theorembdnth 13021 A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
BOUNDED

TheorembdnthALT 13022 Alternate proof of bdnth 13021 not using bdfal 13020. Then, bdfal 13020 can be proved from this theorem, using fal 1338. The total number of proof steps would be 17 (for bdnthALT 13022) + 3 = 20, which is more than 8 (for bdfal 13020) + 9 (for bdnth 13021) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED

Theorembdxor 13023 The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED

Theorembj-bdcel 13024* Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
BOUNDED        BOUNDED

Theorembdab 13025 Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcdeq 13026 Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED CondEq

11.2.7.2  Bounded classes

In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 13028. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas.

As will be clear by the end of this subsection (see for instance bdop 13062), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, BOUNDED BOUNDED . The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like BOUNDED BOUNDED .

Syntaxwbdc 13027 Syntax for the predicate BOUNDED.
BOUNDED

Definitiondf-bdc 13028* Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED BOUNDED

Theorembdceq 13029 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
BOUNDED BOUNDED

Theorembdceqi 13030 A class equal to a bounded one is bounded. Note the use of ax-ext 2119. See also bdceqir 13031. (Contributed by BJ, 3-Oct-2019.)
BOUNDED               BOUNDED

Theorembdceqir 13031 A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13030) equality in the hypothesis, to work better with definitions ( is the definiendum that one wants to prove bounded; see comment of bd0r 13012). (Contributed by BJ, 3-Oct-2019.)
BOUNDED               BOUNDED

Theorembdel 13032* The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED BOUNDED

Theorembdeli 13033* Inference associated with bdel 13032. Its converse is bdelir 13034. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED

Theorembdelir 13034* Inference associated with df-bdc 13028. Its converse is bdeli 13033. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcv 13035 A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
BOUNDED

Theorembdcab 13036 A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
BOUNDED        BOUNDED

Theorembdph 13037 A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
BOUNDED        BOUNDED

Theorembds 13038* Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 13009; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 13009. (Contributed by BJ, 19-Nov-2019.)
BOUNDED               BOUNDED

Theorembdcrab 13039* A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED

Theorembdne 13040 Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdnel 13041* Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

Theorembdreu 13042* Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula need not be bounded even if and are. Indeed, is bounded by bdcvv 13044, and (in minimal propositional calculus), so by bd0 13011, if were bounded when is bounded, then would be bounded as well when is bounded, which is not the case. The same remark holds with . (Contributed by BJ, 16-Oct-2019.)

BOUNDED        BOUNDED

Theorembdrmo 13043* Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcvv 13044 The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.)
BOUNDED

Theorembdsbc 13045 A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13046. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

TheorembdsbcALT 13046 Alternate proof of bdsbc 13045. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED        BOUNDED

Theorembdccsb 13047 A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcdif 13048 The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED

Theorembdcun 13049 The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED

Theorembdcin 13050 The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED        BOUNDED

Theorembdss 13051 The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcnul 13052 The empty class is bounded. See also bdcnulALT 13053. (Contributed by BJ, 3-Oct-2019.)
BOUNDED

TheorembdcnulALT 13053 Alternate proof of bdcnul 13052. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 13031, or use the corresponding characterizations of its elements followed by bdelir 13034. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED

Theorembdeq0 13054 Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED

Theorembj-bd0el 13055 Boundedness of the formula "the empty set belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.)
BOUNDED

Theorembdcpw 13056 The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcsn 13057 The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdcpr 13058 The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdctp 13059 The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdsnss 13060* Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

Theorembdvsn 13061* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdop 13062 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED

Theorembdcuni 13063 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
BOUNDED

Theorembdcint 13064 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdciun 13065* The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

Theorembdciin 13066* The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED        BOUNDED

Theorembdcsuc 13067 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED

Theorembdeqsuc 13068* Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
BOUNDED

Theorembj-bdsucel 13069 Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.)
BOUNDED

Theorembdcriota 13070* A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
BOUNDED               BOUNDED

11.2.8  CZF: Bounded separation

In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.

Axiomax-bdsep 13071* Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 4041. (Contributed by BJ, 5-Oct-2019.)
BOUNDED

Theorembdsep1 13072* Version of ax-bdsep 13071 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
BOUNDED

Theorembdsep2 13073* Version of ax-bdsep 13071 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13072 when sufficient. (Contributed by BJ, 5-Oct-2019.)
BOUNDED

Theorembdsepnft 13074* Closed form of bdsepnf 13075. Version of ax-bdsep 13071 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 13072 when sufficient. (Contributed by BJ, 19-Oct-2019.)
BOUNDED

Theorembdsepnf 13075* Version of ax-bdsep 13071 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13076. Use bdsep1 13072 when sufficient. (Contributed by BJ, 5-Oct-2019.)
BOUNDED

TheorembdsepnfALT 13076* Alternate proof of bdsepnf 13075, not using bdsepnft 13074. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED

Theorembdzfauscl 13077* Closed form of the version of zfauscl 4043 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
BOUNDED

Theorembdbm1.3ii 13078* Bounded version of bm1.3ii 4044. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembj-axemptylem 13079* Lemma for bj-axempty 13080 and bj-axempty2 13081. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4049 instead. (New usage is discouraged.)

Theorembj-axempty 13080* Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4048. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4049 instead. (New usage is discouraged.)

Theorembj-axempty2 13081* Axiom of the empty set from bounded separation, alternate version to bj-axempty 13080. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4049 instead. (New usage is discouraged.)

Theorembj-nalset 13082* nalset 4053 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Theorembj-vprc 13083 vprc 4055 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Theorembj-nvel 13084 nvel 4056 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Theorembj-vnex 13085 vnex 4054 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Theorembdinex1 13086 Bounded version of inex1 4057. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdinex2 13087 Bounded version of inex2 4058. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdinex1g 13088 Bounded version of inex1g 4059. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdssex 13089 Bounded version of ssex 4060. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdssexi 13090 Bounded version of ssexi 4061. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdssexg 13091 Bounded version of ssexg 4062. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdssexd 13092 Bounded version of ssexd 4063. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED

Theorembdrabexg 13093* Bounded version of rabexg 4066. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED        BOUNDED

Theorembj-inex 13094 The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Theorembj-intexr 13095 intexr 4070 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Theorembj-intnexr 13096 intnexr 4071 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)

Theorembj-zfpair2 13097 Proof of zfpair2 4127 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Theorembj-prexg 13098 Proof of prexg 4128 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Theorembj-snexg 13099 snexg 4103 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Theorembj-snex 13100 snex 4104 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

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 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13239
 Copyright terms: Public domain < Previous  Next >