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Type | Label | Description |
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Statement | ||
Theorem | bj-elssuniab 10601 | Version of elssuni 3629 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) |
Theorem | bj-sseq 10602 | If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.) |
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 3896 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 10675. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 3893 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 10777 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 10736. Similarly, the axiom of powerset ax-pow 3948 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 10782. 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 4280. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 10763. 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) 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) 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 intuitonistic, 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 10604. 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 10604 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 10605 through ax-bdsb 10613) can be written either in closed or inference form. The fact that ax-bd0 10604 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 10612. For a similar method, see bj-omtrans 10751. Note that one cannot add an axiom BOUNDED since by bdph 10641 it would imply that every formula is bounded. | ||
Syntax | wbd 10603 | Syntax for the predicate BOUNDED. |
BOUNDED | ||
Axiom | ax-bd0 10604 | 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 10605 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdan 10606 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdor 10607 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdn 10608 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdal 10609* | A bounded universal quantification of a bounded formula is bounded. Note the DV condition on . (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdex 10610* | A bounded existential quantification of a bounded formula is bounded. Note the DV condition on . (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdeq 10611 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Axiom | ax-bdel 10612 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Axiom | ax-bdsb 10613 | 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 1686, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdeq 10614 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bd0 10615 | A formula equivalent to a bounded one is bounded. See also bd0r 10616. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bd0r 10616 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 10615) 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 10617 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdstab 10618 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED STAB | ||
Theorem | bddc 10619 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED DECID | ||
Theorem | bd3or 10620 | A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED BOUNDED | ||
Theorem | bd3an 10621 | A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED BOUNDED | ||
Theorem | bdth 10622 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED | ||
Theorem | bdtru 10623 | The truth value is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdfal 10624 | The truth value is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdnth 10625 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED | ||
Theorem | bdnthALT 10626 | Alternate proof of bdnth 10625 not using bdfal 10624. Then, bdfal 10624 can be proved from this theorem, using fal 1291. The total number of proof steps would be 17 (for bdnthALT 10626) + 3 = 20, which is more than 8 (for bdfal 10624) + 9 (for bdnth 10625) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdxor 10627 | The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bj-bdcel 10628* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdab 10629 | 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 10630 | 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 10632. 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 10666), 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 10631 | Syntax for the predicate BOUNDED. |
BOUNDED | ||
Definition | df-bdc 10632* | 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 10633 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceqi 10634 | A class equal to a bounded one is bounded. Note the use of ax-ext 2063. See also bdceqir 10635. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceqir 10635 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 10634) equality in the hypothesis, to work better with definitions ( is the definiendum that one wants to prove bounded; see comment of bd0r 10616). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdel 10636* | The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdeli 10637* | Inference associated with bdel 10636. Its converse is bdelir 10638. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdelir 10638* | Inference associated with df-bdc 10632. Its converse is bdeli 10637. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcv 10639 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdcab 10640 | A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdph 10641 | A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bds 10642* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10613; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10613. (Contributed by BJ, 19-Nov-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcrab 10643* | 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 10644 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdnel 10645* | Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdreu 10646* |
Boundedness of existential uniqueness.
Remark regarding restricted quantifiers: the formula need not be bounded even if and are. Indeed, is bounded by bdcvv 10648, and (in minimal propositional calculus), so by bd0 10615, 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 10647* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcvv 10648 | 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 10649 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 10650. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdsbcALT 10650 | Alternate proof of bdsbc 10649. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bdccsb 10651 | 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 10652 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdcun 10653 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdcin 10654 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdss 10655 | 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 10656 | The empty class is bounded. See also bdcnulALT 10657. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdcnulALT 10657 | Alternate proof of bdcnul 10656. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 10635, or use the corresponding characterizations of its elements followed by bdelir 10638. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdeq0 10658 | Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bj-bd0el 10659 | Boundedness of the formula "the empty set belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcpw 10660 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcsn 10661 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdcpr 10662 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdctp 10663 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdsnss 10664* | 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 10665* | Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdop 10666 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bdcuni 10667 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
BOUNDED | ||
Theorem | bdcint 10668 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdciun 10669* | 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 10670* | 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 10671 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdeqsuc 10672* | 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 10673 | Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcriota 10674* | 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 10675* | 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 3896. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep1 10676* | Version of ax-bdsep 10675 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep2 10677* | Version of ax-bdsep 10675 with one DV condition removed and without initial universal quantifier. Use bdsep1 10676 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnft 10678* | Closed form of bdsepnf 10679. Version of ax-bdsep 10675 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10676 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnf 10679* | Version of ax-bdsep 10675 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10680. Use bdsep1 10676 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnfALT 10680* | Alternate proof of bdsepnf 10679, not using bdsepnft 10678. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdzfauscl 10681* | Closed form of the version of zfauscl 3898 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
BOUNDED | ||
Theorem | bdbm1.3ii 10682* | Bounded version of bm1.3ii 3899. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-axemptylem 10683* | Lemma for bj-axempty 10684 and bj-axempty2 10685. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.) |
Theorem | bj-axempty 10684* | Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3903. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.) |
Theorem | bj-axempty2 10685* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 10684. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.) |
Theorem | bj-nalset 10686* | nalset 3908 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vprc 10687 | vprc 3909 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nvel 10688 | nvel 3910 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vnex 10689 | vnex 3911 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdinex1 10690 | Bounded version of inex1 3912. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex2 10691 | Bounded version of inex2 3913. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex1g 10692 | Bounded version of inex1g 3914. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssex 10693 | Bounded version of ssex 3915. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexi 10694 | Bounded version of ssexi 3916. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexg 10695 | Bounded version of ssexg 3917. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexd 10696 | Bounded version of ssexd 3918. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdrabexg 10697* | Bounded version of rabexg 3921. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bj-inex 10698 | The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intexr 10699 | intexr 3925 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intnexr 10700 | intnexr 3926 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
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