P. 154 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15301-15400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-charfundcALT 15301*	Alternate proof of bj-charfundc 15300. It was expected to be much shorter since it uses bj-charfun 15299 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.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfunr 15302*	If a class 𝐴 has a "weak" characteristic function on a class 𝑋, then negated membership in 𝐴 is decidable (in other words, membership in 𝐴 is testable) in 𝑋. The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as ⊢ (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful. The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)
Theorem	bj-charfunbi 15303*	In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function. This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
13.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes 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 4147 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 15376. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4144 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 15474 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 15433. Similarly, the axiom of powerset ax-pow 4203 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 15479. 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 4569. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 15460. 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 15460) 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 15460 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
13.2.8.1  Bounded formulas 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 𝜑 " is a formula meaning that 𝜑 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, ∀𝑥⊤ 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 15305. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (𝜑 → BOUNDED 𝜑) and ⊢ (¬ 𝜑 → BOUNDED 𝜑) 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 15305 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 15306 through ax-bdsb 15314) can be written either in closed or inference form. The fact that ax-bd0 15305 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 𝑥 ∈ ω is a bounded formula. However, since ω can be defined as "the 𝑦 such that PHI" a proof using the fact that 𝑥 ∈ ω is bounded can be converted to a proof in iset.mm by replacing ω with 𝑦 everywhere and prepending the antecedent PHI, since 𝑥 ∈ 𝑦 is bounded by ax-bdel 15313. For a similar method, see bj-omtrans 15448. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 15342 it would imply that every formula is bounded.
Syntax	wbd 15304	Syntax for the predicate BOUNDED.
wff BOUNDED 𝜑
Axiom	ax-bd0 15305	If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
Axiom	ax-bdim 15306	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 → 𝜓)
Axiom	ax-bdan 15307	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓)
Axiom	ax-bdor 15308	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓)
Axiom	ax-bdn 15309	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ¬ 𝜑
Axiom	ax-bdal 15310*	A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdex 15311*	A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdeq 15312	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 = 𝑦
Axiom	ax-bdel 15313	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝑦
Axiom	ax-bdsb 15314	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 1774, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdeq 15315	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)
Theorem	bd0 15316	A formula equivalent to a bounded one is bounded. See also bd0r 15317. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bd0r 15317	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 15316) biconditional in the hypothesis, to work better with definitions (𝜓 is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdbi 15318	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ↔ 𝜓)
Theorem	bdstab 15319	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED STAB 𝜑
Theorem	bddc 15320	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED DECID 𝜑
Theorem	bd3or 15321	A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓 ∨ 𝜒)
Theorem	bd3an 15322	A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒)
Theorem	bdth 15323	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdtru 15324	The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊤
Theorem	bdfal 15325	The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊥
Theorem	bdnth 15326	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdnthALT 15327	Alternate proof of bdnth 15326 not using bdfal 15325. Then, bdfal 15325 can be proved from this theorem, using fal 1371. The total number of proof steps would be 17 (for bdnthALT 15327) + 3 = 20, which is more than 8 (for bdfal 15325) + 9 (for bdnth 15326) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdxor 15328	The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ⊻ 𝜓)
Theorem	bj-bdcel 15329*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
⊢ BOUNDED 𝑦 = 𝐴    ⇒   ⊢ BOUNDED 𝐴 ∈ 𝑥
Theorem	bdab 15330	Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}
Theorem	bdcdeq 15331	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑)
13.2.8.2  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 15333. 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 15367), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, ⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ⟨{𝑥 ∣ 𝜑}, ({𝑦, suc 𝑧} × ⟨𝑡, ∅⟩)⟩. 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 𝐴 ⇒ ⊢ BOUNDED {𝐴}.
Syntax	wbdc 15332	Syntax for the predicate BOUNDED.
wff BOUNDED 𝐴
Definition	df-bdc 15333*	Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdceq 15334	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
Theorem	bdceqi 15335	A class equal to a bounded one is bounded. Note the use of ax-ext 2175. See also bdceqir 15336. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdceqir 15336	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 15335) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 15317). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 = 𝐴    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdel 15337*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdeli 15338*	Inference associated with bdel 15337. Its converse is bdelir 15339. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∈ 𝐴
Theorem	bdelir 15339*	Inference associated with df-bdc 15333. Its converse is bdeli 15338. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝐴    ⇒   ⊢ BOUNDED 𝐴
Theorem	bdcv 15340	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥
Theorem	bdcab 15341	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∣ 𝜑}
Theorem	bdph 15342	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED {𝑥 ∣ 𝜑}    ⇒   ⊢ BOUNDED 𝜑
Theorem	bds 15343*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 15314; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 15314. (Contributed by BJ, 19-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdcrab 15344*	A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bdne 15345	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 ≠ 𝑦
Theorem	bdnel 15346*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∉ 𝐴
Theorem	bdreu 15347*	Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 15349, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 15316, if ∀𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then ∀𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
Theorem	bdrmo 15348*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑
Theorem	bdcvv 15349	The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED V
Theorem	bdsbc 15350	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 15351. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdsbcALT 15351	Alternate proof of bdsbc 15350. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdccsb 15352	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 𝐴    ⇒   ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴
Theorem	bdcdif 15353	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∖ 𝐵)
Theorem	bdcun 15354	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∪ 𝐵)
Theorem	bdcin 15355	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∩ 𝐵)
Theorem	bdss 15356	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 𝐴    ⇒   ⊢ BOUNDED 𝑥 ⊆ 𝐴
Theorem	bdcnul 15357	The empty class is bounded. See also bdcnulALT 15358. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ∅
Theorem	bdcnulALT 15358	Alternate proof of bdcnul 15357. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 15336, or use the corresponding characterizations of its elements followed by bdelir 15339. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED ∅
Theorem	bdeq0 15359	Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED 𝑥 = ∅
Theorem	bj-bd0el 15360	Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED ∅ ∈ 𝑥
Theorem	bdcpw 15361	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝒫 𝐴
Theorem	bdcsn 15362	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥}
Theorem	bdcpr 15363	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦}
Theorem	bdctp 15364	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦, 𝑧}
Theorem	bdsnss 15365*	Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED {𝑥} ⊆ 𝐴
Theorem	bdvsn 15366*	Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 = {𝑦}
Theorem	bdop 15367	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED ⟨𝑥, 𝑦⟩
Theorem	bdcuni 15368	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
⊢ BOUNDED ∪ 𝑥
Theorem	bdcint 15369	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED ∩ 𝑥
Theorem	bdciun 15370*	The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴
Theorem	bdciin 15371*	The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴
Theorem	bdcsuc 15372	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED suc 𝑥
Theorem	bdeqsuc 15373*	Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED 𝑥 = suc 𝑦
Theorem	bj-bdsucel 15374	Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED suc 𝑥 ∈ 𝑦
Theorem	bdcriota 15375*	A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ ∃!𝑥 ∈ 𝑦 𝜑    ⇒   ⊢ BOUNDED (℩𝑥 ∈ 𝑦 𝜑)
13.2.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 15376*	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 4147. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep1 15377*	Version of ax-bdsep 15376 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep2 15378*	Version of ax-bdsep 15376 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 15377 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnft 15379*	Closed form of bdsepnf 15380. Version of ax-bdsep 15376 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 15377 when sufficient. (Contributed by BJ, 19-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
Theorem	bdsepnf 15380*	Version of ax-bdsep 15376 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 15381. Use bdsep1 15377 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ Ⅎ𝑏𝜑    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnfALT 15381*	Alternate proof of bdsepnf 15380, not using bdsepnft 15379. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑏𝜑    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdzfauscl 15382*	Closed form of the version of zfauscl 4149 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	bdbm1.3ii 15383*	Bounded version of bm1.3ii 4150. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)    ⇒   ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
Theorem	bj-axemptylem 15384*	Lemma for bj-axempty 15385 and bj-axempty2 15386. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4155 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
Theorem	bj-axempty 15385*	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 4154. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4155 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥
Theorem	bj-axempty2 15386*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 15385. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4155 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	bj-nalset 15387*	nalset 4159 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	bj-vprc 15388	vprc 4161 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ V
Theorem	bj-nvel 15389	nvel 4162 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	bj-vnex 15390	vnex 4160 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	bdinex1 15391	Bounded version of inex1 4163. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ V
Theorem	bdinex2 15392	Bounded version of inex2 4164. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	bdinex1g 15393	Bounded version of inex1g 4165. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bdssex 15394	Bounded version of ssex 4166. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	bdssexi 15395	Bounded version of ssexi 4167. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	bdssexg 15396	Bounded version of ssexg 4168. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	bdssexd 15397	Bounded version of ssexd 4169. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	bdrabexg 15398*	Bounded version of rabexg 4172. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	bj-inex 15399	The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bj-intexr 15400	intexr 4179 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)