![]() |
Intuitionistic Logic Explorer Theorem List (p. 143 of 145) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
The question of decidability is essential in intuitionistic logic. In
intuitionistic set theories, it is natural to define decidability of a set
(or class) as decidability of membership in it. One can parameterize this
notion with another set (or class) since it is often important to assess
decidability of membership in one class among elements of another class.
Namely, one will say that " Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 14249). | ||
Syntax | wdcin 14201 | Syntax for decidability of a class in another. |
![]() ![]() ![]() | ||
Definition | df-dcin 14202* | Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | decidi 14203 | Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | decidr 14204* | Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | decidin 14205 | 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | uzdcinzz 14206 | An upperset of integers is decidable in the integers. Reformulation of eluzdc 9599. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | sumdc2 14207* |
Alternate proof of sumdc 11350, without disjoint variable condition on
![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djucllem 14208* | Lemma for djulcl 7044 and djurcl 7045. (Contributed by BJ, 4-Jul-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djulclALT 14209 | Shortening of djulcl 7044 using djucllem 14208. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djurclALT 14210 | Shortening of djurcl 7045 using djucllem 14208. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | funmptd 14211 |
The maps-to notation defines a function (deduction form).
Note: one should similarly prove a deduction form of funopab4 5249, then prove funmptd 14211 from it, and then prove funmpt 5250 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fnmptd 14212* | The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | if0ab 14213* |
Expression of a conditional class as a class abstraction when the False
alternative is the empty class: in that case, the conditional class is
the extension, in the True alternative, of the condition.
Remark: a consequence which could be formalized is the inclusion
|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fmelpw1o 14214 |
With a formula ![]() ![]() ![]() ![]() ![]() ![]() ![]()
As proved in if0ab 14213, the associated element of |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-charfun 14215* |
Properties of the characteristic function on the class ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-charfundc 14216* |
Properties of the characteristic function on the class ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-charfundcALT 14217* | Alternate proof of bj-charfundc 14216. It was expected to be much shorter since it uses bj-charfun 14215 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-charfunr 14218* |
If a class ![]() ![]() ![]() ![]() ![]()
The hypothesis imposes that
The theorem would still hold if the codomain of |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-charfunbi 14219* |
In an ambient set ![]() ![]() ![]()
This characterization can be applied to singletons when the set |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
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 4118 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 14292. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4115 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 14390 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 14349. Similarly, the axiom of powerset ax-pow 4171 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 14395. 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 4533. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 14376. 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 14376) 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. https://doi.org/10.48550/arXiv.1808.05204 14376 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
A second choice is to view "bounded" either as a syntactic or a
semantic
property.
For instance,
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 14221.
Indeed, if we posited it in closed form, then we could prove for instance
Having ax-bd0 14221 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 14222 through ax-bdsb 14230) can be written either in closed or inference form. The fact that ax-bd0 14221 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
Note that one cannot add an axiom | ||
Syntax | wbd 14220 | Syntax for the predicate BOUNDED. |
![]() ![]() | ||
Axiom | ax-bd0 14221 | If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdim 14222 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdan 14223 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdor 14224 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdn 14225 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdal 14226* |
A bounded universal quantification of a bounded formula is bounded.
Note the disjoint variable condition on ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdex 14227* |
A bounded existential quantification of a bounded formula is bounded.
Note the disjoint variable condition on ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Axiom | ax-bdeq 14228 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() | ||
Axiom | ax-bdel 14229 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() | ||
Axiom | ax-bdsb 14230 | 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 1763, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdeq 14231 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bd0 14232 | A formula equivalent to a bounded one is bounded. See also bd0r 14233. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bd0r 14233 |
A formula equivalent to a bounded one is bounded. Stated with a
commuted (compared with bd0 14232) biconditional in the hypothesis, to work
better with definitions (![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdbi 14234 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdstab 14235 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() | ||
Theorem | bddc 14236 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() | ||
Theorem | bd3or 14237 | A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bd3an 14238 | A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdth 14239 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdtru 14240 |
The truth value ![]() |
![]() ![]() | ||
Theorem | bdfal 14241 |
The truth value ![]() |
![]() ![]() | ||
Theorem | bdnth 14242 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdnthALT 14243 | Alternate proof of bdnth 14242 not using bdfal 14241. Then, bdfal 14241 can be proved from this theorem, using fal 1360. The total number of proof steps would be 17 (for bdnthALT 14243) + 3 = 20, which is more than 8 (for bdfal 14241) + 9 (for bdnth 14242) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdxor 14244 | The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-bdcel 14245* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdab 14246 | Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcdeq 14247 | Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
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 14249. 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 14283),
one can prove the boundedness of any concrete term using only setvars and
bounded formulas, for instance,
| ||
Syntax | wbdc 14248 | Syntax for the predicate BOUNDED. |
![]() ![]() | ||
Definition | df-bdc 14249* | Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdceq 14250 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdceqi 14251 | A class equal to a bounded one is bounded. Note the use of ax-ext 2159. See also bdceqir 14252. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdceqir 14252 |
A class equal to a bounded one is bounded. Stated with a commuted
(compared with bdceqi 14251) equality in the hypothesis, to work better
with definitions (![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdel 14253* | The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdeli 14254* | Inference associated with bdel 14253. Its converse is bdelir 14255. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdelir 14255* | Inference associated with df-bdc 14249. Its converse is bdeli 14254. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcv 14256 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() | ||
Theorem | bdcab 14257 | A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdph 14258 | A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bds 14259* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 14230; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 14230. (Contributed by BJ, 19-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcrab 14260* | A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdne 14261 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() | ||
Theorem | bdnel 14262* | Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdreu 14263* |
Boundedness of existential uniqueness.
Remark regarding restricted quantifiers: the formula |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdrmo 14264* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcvv 14265 | The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() | ||
Theorem | bdsbc 14266 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 14267. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsbcALT 14267 | Alternate proof of bdsbc 14266. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdccsb 14268 | A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcdif 14269 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcun 14270 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcin 14271 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdss 14272 | 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.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcnul 14273 | The empty class is bounded. See also bdcnulALT 14274. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() | ||
Theorem | bdcnulALT 14274 | Alternate proof of bdcnul 14273. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 14252, or use the corresponding characterizations of its elements followed by bdelir 14255. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() | ||
Theorem | bdeq0 14275 | Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) |
![]() ![]() ![]() ![]() | ||
Theorem | bj-bd0el 14276 |
Boundedness of the formula "the empty set belongs to the setvar ![]() |
![]() ![]() ![]() ![]() | ||
Theorem | bdcpw 14277 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcsn 14278 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() | ||
Theorem | bdcpr 14279 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdctp 14280 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsnss 14281* | Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdvsn 14282* | Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdop 14283 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcuni 14284 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
![]() ![]() ![]() | ||
Theorem | bdcint 14285 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() | ||
Theorem | bdciun 14286* | The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdciin 14287* | The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcsuc 14288 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
![]() ![]() ![]() | ||
Theorem | bdeqsuc 14289* | Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-bdsucel 14290 |
Boundedness of the formula "the successor of the setvar ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdcriota 14291* | A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory. | ||
Axiom | ax-bdsep 14292* | 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 4118. (Contributed by BJ, 5-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsep1 14293* | Version of ax-bdsep 14292 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsep2 14294* | Version of ax-bdsep 14292 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 14293 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsepnft 14295* | Closed form of bdsepnf 14296. Version of ax-bdsep 14292 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 14293 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsepnf 14296* | Version of ax-bdsep 14292 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 14297. Use bdsep1 14293 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdsepnfALT 14297* | Alternate proof of bdsepnf 14296, not using bdsepnft 14295. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdzfauscl 14298* | Closed form of the version of zfauscl 4120 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bdbm1.3ii 14299* | Bounded version of bm1.3ii 4121. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bj-axemptylem 14300* | Lemma for bj-axempty 14301 and bj-axempty2 14302. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4126 instead. (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |