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Type | Label | Description |
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Statement | ||
Definition | df-dcin 13001* | Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.) |
DECIDin DECID | ||
Theorem | decidi 13002 | Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.) |
DECIDin | ||
Theorem | decidr 13003* | Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.) |
DECIDin | ||
Theorem | decidin 13004 | If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.) |
DECIDin DECIDin DECIDin | ||
Theorem | uzdcinzz 13005 | An upperset of integers is decidable in the integers. Reformulation of eluzdc 9404. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.) |
DECIDin | ||
Theorem | sumdc2 13006* | Alternate proof of sumdc 11127, without disjoint variable condition on (longer because the statement is taylored to the proof sumdc 11127). (Contributed by BJ, 19-Feb-2022.) |
DECID DECID | ||
Theorem | djucllem 13007* | Lemma for djulcl 6936 and djurcl 6937. (Contributed by BJ, 4-Jul-2022.) |
Theorem | djulclALT 13008 | Shortening of djulcl 6936 using djucllem 13007. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
inl ⊔ | ||
Theorem | djurclALT 13009 | Shortening of djurcl 6937 using djucllem 13007. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
inr ⊔ | ||
This section develops constructive Zermelo--Fraenkel set theory (CZF) on top of intuitionistic logic. It is a constructive theory in the sense that its logic is intuitionistic and it is predicative. "Predicative" means that new sets can be constructed only from already constructed sets. In particular, the axiom of separation ax-sep 4046 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 13082. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4043 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 13180 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 13139. Similarly, the axiom of powerset ax-pow 4098 is not predicative (checking whether a set is included in another requires to universally quantifier over that "not yet constructed" set) and is replaced in CZF by the axiom of fullness or the axiom of subset collection ax-sscoll 13185. In an intuitionistic context, the axiom of regularity is stated in IZF as well as in CZF as the axiom of set induction ax-setind 4452. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 13166. For more details on CZF, a useful set of notes is Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf 13166) and an interesting article is Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. (available at https://arxiv.org/abs/1808.05204 13166) I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results. | ||
The present definition of bounded formulas emerged from a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein). In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ0) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ0) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitionistic, for instance to state the axiom scheme of Δ0-induction. To formalize this in Metamath, there are several choices to make. A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph0 ...) and an axiom "$a wff ph0 " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph0 -> ps0 )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED " is a formula meaning that is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.) A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded. A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 13011. Indeed, if we posited it in closed form, then we could prove for instance BOUNDED and BOUNDED which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.) Having ax-bd0 13011 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 13012 through ax-bdsb 13020) can be written either in closed or inference form. The fact that ax-bd0 13011 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness. Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that is a bounded formula. However, since can be defined as "the such that PHI" a proof using the fact that is bounded can be converted to a proof in iset.mm by replacing with everywhere and prepending the antecedent PHI, since is bounded by ax-bdel 13019. For a similar method, see bj-omtrans 13154. Note that one cannot add an axiom BOUNDED since by bdph 13048 it would imply that every formula is bounded. | ||
Syntax | wbd 13010 | Syntax for the predicate BOUNDED. |
BOUNDED | ||
Axiom | ax-bd0 13011 | If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdim 13012 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdan 13013 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdor 13014 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdn 13015 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdal 13016* | A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on . (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdex 13017* | A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on . (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdeq 13018 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Axiom | ax-bdel 13019 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Axiom | ax-bdsb 13020 | 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 | ||
Theorem | bdeq 13021 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bd0 13022 | A formula equivalent to a bounded one is bounded. See also bd0r 13023. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bd0r 13023 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13022) 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 | ||
Theorem | bdbi 13024 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdstab 13025 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED STAB | ||
Theorem | bddc 13026 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED DECID | ||
Theorem | bd3or 13027 | A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED BOUNDED | ||
Theorem | bd3an 13028 | A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED BOUNDED | ||
Theorem | bdth 13029 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED | ||
Theorem | bdtru 13030 | The truth value is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdfal 13031 | The truth value is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdnth 13032 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED | ||
Theorem | bdnthALT 13033 | Alternate proof of bdnth 13032 not using bdfal 13031. Then, bdfal 13031 can be proved from this theorem, using fal 1338. The total number of proof steps would be 17 (for bdnthALT 13033) + 3 = 20, which is more than 8 (for bdfal 13031) + 9 (for bdnth 13032) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdxor 13034 | The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bj-bdcel 13035* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdab 13036 | Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcdeq 13037 | Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED CondEq | ||
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 13039. 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 13073), 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 . | ||
Syntax | wbdc 13038 | Syntax for the predicate BOUNDED. |
BOUNDED | ||
Definition | df-bdc 13039* | Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceq 13040 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceqi 13041 | A class equal to a bounded one is bounded. Note the use of ax-ext 2121. See also bdceqir 13042. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceqir 13042 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13041) equality in the hypothesis, to work better with definitions ( is the definiendum that one wants to prove bounded; see comment of bd0r 13023). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdel 13043* | The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdeli 13044* | Inference associated with bdel 13043. Its converse is bdelir 13045. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdelir 13045* | Inference associated with df-bdc 13039. Its converse is bdeli 13044. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcv 13046 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdcab 13047 | A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdph 13048 | A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bds 13049* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 13020; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 13020. (Contributed by BJ, 19-Nov-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcrab 13050* | 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 | ||
Theorem | bdne 13051 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdnel 13052* | Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdreu 13053* |
Boundedness of existential uniqueness.
Remark regarding restricted quantifiers: the formula need not be bounded even if and are. Indeed, is bounded by bdcvv 13055, and (in minimal propositional calculus), so by bd0 13022, 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 | ||
Theorem | bdrmo 13054* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcvv 13055 | The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdsbc 13056 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13057. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdsbcALT 13057 | Alternate proof of bdsbc 13056. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bdccsb 13058 | 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 | ||
Theorem | bdcdif 13059 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdcun 13060 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdcin 13061 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdss 13062 | 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 | ||
Theorem | bdcnul 13063 | The empty class is bounded. See also bdcnulALT 13064. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdcnulALT 13064 | Alternate proof of bdcnul 13063. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 13042, or use the corresponding characterizations of its elements followed by bdelir 13045. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdeq0 13065 | Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bj-bd0el 13066 | Boundedness of the formula "the empty set belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcpw 13067 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcsn 13068 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdcpr 13069 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdctp 13070 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdsnss 13071* | Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdvsn 13072* | Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdop 13073 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bdcuni 13074 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
BOUNDED | ||
Theorem | bdcint 13075 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdciun 13076* | The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdciin 13077* | The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcsuc 13078 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdeqsuc 13079* | Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bj-bdsucel 13080 | Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcriota 13081* | A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.) |
BOUNDED BOUNDED | ||
In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory. | ||
Axiom | ax-bdsep 13082* | 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 4046. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep1 13083* | Version of ax-bdsep 13082 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep2 13084* | Version of ax-bdsep 13082 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13083 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnft 13085* | Closed form of bdsepnf 13086. Version of ax-bdsep 13082 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 13083 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnf 13086* | Version of ax-bdsep 13082 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 13087. Use bdsep1 13083 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnfALT 13087* | Alternate proof of bdsepnf 13086, not using bdsepnft 13085. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdzfauscl 13088* | Closed form of the version of zfauscl 4048 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
BOUNDED | ||
Theorem | bdbm1.3ii 13089* | Bounded version of bm1.3ii 4049. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-axemptylem 13090* | Lemma for bj-axempty 13091 and bj-axempty2 13092. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Theorem | bj-axempty 13091* | 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 4053. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Theorem | bj-axempty2 13092* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 13091. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Theorem | bj-nalset 13093* | nalset 4058 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vprc 13094 | vprc 4060 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nvel 13095 | nvel 4061 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vnex 13096 | vnex 4059 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdinex1 13097 | Bounded version of inex1 4062. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex2 13098 | Bounded version of inex2 4063. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex1g 13099 | Bounded version of inex1g 4064. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssex 13100 | Bounded version of ssex 4065. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED |
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