P. 164 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16301-16400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	decidi 16301	Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
|- ( A DECIDin B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )
Theorem	decidr 16302*	Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
|- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )   =>    |- ( ph -> A DECIDin B )
Theorem	decidin 16303	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.)
|- ( ph -> A C_ B )   &    |- ( ph -> A DECIDin B )   &    |- ( ph -> B DECIDin C )   =>    |- ( ph -> A DECIDin C )
Theorem	uzdcinzz 16304	An upperset of integers is decidable in the integers. Reformulation of eluzdc 9832. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
|- ( M e. ZZ -> ( ZZ>= ` M ) DECIDin ZZ )
Theorem	sumdc2 16305*	Alternate proof of sumdc 11906, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11906). (Contributed by BJ, 19-Feb-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. x e. ( ZZ>= ` M )DECID x e. A )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> DECID N e. A )
14.2.6  Disjoint union
Theorem	djucllem 16306*	Lemma for djulcl 7239 and djurcl 7240. (Contributed by BJ, 4-Jul-2022.)
|- X e. _V   &    |- F = ( x e. _V |-> <. X , x >. )   =>    |- ( A e. B -> ( ( F |` B ) ` A ) e. ( { X } X. B ) )
Theorem	djulclALT 16307	Shortening of djulcl 7239 using djucllem 16306. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )
Theorem	djurclALT 16308	Shortening of djurcl 7240 using djucllem 16306. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )
14.2.7  Miscellaneous
Theorem	funmptd 16309	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5359, then prove funmptd 16309 from it, and then prove funmpt 5360 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)
|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ph -> Fun F )
Theorem	fnmptd 16310*	The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ph -> F Fn A )
Theorem	if0ab 16311*	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 |- if ( ph , A , (/) ) C_ A and therefore, using elpwg 3658, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 7453 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
|- if ( ph , A , (/) ) = { x e. A | ph }
Theorem	bj-charfun 16312*	Properties of the characteristic function on the class X of the class A. (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   =>    |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
Theorem	bj-charfundc 16313*	Properties of the characteristic function on the class X of the class A, provided membership in A is decidable in X. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
Theorem	bj-charfundcALT 16314*	Alternate proof of bj-charfundc 16313. It was expected to be much shorter since it uses bj-charfun 16312 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.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
Theorem	bj-charfunr 16315*	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 |- ( ph -> ( F : X --> om /\ ... ) ) 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.)
|- ( ph -> E. f e. ( om ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) =/= (/) /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) )   =>    |- ( ph -> A. x e. X DECID -. x e. A )
Theorem	bj-charfunbi 16316*	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.)
|- ( ph -> X e. V )   &    |- ( ph -> A. x e. X STAB x e. A )   =>    |- ( ph -> ( A. x e. X DECID x e. A <-> E. f e. ( 2o ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) = 1o /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) ) )
14.2.8  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 4203 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 16389. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4200 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 16487 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 16446. Similarly, the axiom of powerset ax-pow 4260 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 16492. 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 4631. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 16473. 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 16473) 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 16473 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
14.2.8.1  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 ph " is a formula meaning that ph 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, A. x T. is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to T. 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 16318. Indeed, if we posited it in closed form, then we could prove for instance |- ( ph -> BOUNDED ph ) and |- ( -. ph -> BOUNDED ph ) 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 16318 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 16319 through ax-bdsb 16327) can be written either in closed or inference form. The fact that ax-bd0 16318 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 e. om is a bounded formula. However, since om can be defined as "the y such that PHI" a proof using the fact that x e. om is bounded can be converted to a proof in iset.mm by replacing om with y everywhere and prepending the antecedent PHI, since x e. y is bounded by ax-bdel 16326. For a similar method, see bj-omtrans 16461. Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 16355 it would imply that every formula is bounded.
Syntax	wbd 16317	Syntax for the predicate BOUNDED.
wff BOUNDED ph
Axiom	ax-bd0 16318	If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph -> BOUNDED ps )
Axiom	ax-bdim 16319	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph -> ps )
Axiom	ax-bdan 16320	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )
Axiom	ax-bdor 16321	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )
Axiom	ax-bdn 16322	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED -. ph
Axiom	ax-bdal 16323*	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 ph   =>    |- BOUNDED A. x e. y ph
Axiom	ax-bdex 16324*	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 ph   =>    |- BOUNDED E. x e. y ph
Axiom	ax-bdeq 16325	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x = y
Axiom	ax-bdel 16326	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. y
Axiom	ax-bdsb 16327	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 1809, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [ y / x ] ph
Theorem	bdeq 16328	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )
Theorem	bd0 16329	A formula equivalent to a bounded one is bounded. See also bd0r 16330. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps
Theorem	bd0r 16330	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16329) biconditional in the hypothesis, to work better with definitions ( ps is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ps <-> ph )   =>    |- BOUNDED ps
Theorem	bdbi 16331	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )
Theorem	bdstab 16332	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED STAB ph
Theorem	bddc 16333	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED DECID ph
Theorem	bd3or 16334	A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   &    |- BOUNDED ch   =>    |- BOUNDED ( ph \/ ps \/ ch )
Theorem	bd3an 16335	A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   &    |- BOUNDED ch   =>    |- BOUNDED ( ph /\ ps /\ ch )
Theorem	bdth 16336	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- ph   =>    |- BOUNDED ph
Theorem	bdtru 16337	The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED T.
Theorem	bdfal 16338	The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED F.
Theorem	bdnth 16339	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdnthALT 16340	Alternate proof of bdnth 16339 not using bdfal 16338. Then, bdfal 16338 can be proved from this theorem, using fal 1402. The total number of proof steps would be 17 (for bdnthALT 16340) + 3 = 20, which is more than 8 (for bdfal 16338) + 9 (for bdnth 16339) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdxor 16341	The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/_ ps )
Theorem	bj-bdcel 16342*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
|- BOUNDED y = A   =>    |- BOUNDED A e. x
Theorem	bdab 16343	Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED x e. { y | ph }
Theorem	bdcdeq 16344	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED CondEq ( x = y -> ph )
14.2.8.2  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 16346. 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 16380), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, |- BOUNDED ph => |- BOUNDED <. { x | ph } , ( { y , suc z } X. <. 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 16345	Syntax for the predicate BOUNDED.
wff BOUNDED A
Definition	df-bdc 16346*	Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- (BOUNDED A <-> A. xBOUNDED x e. A )
Theorem	bdceq 16347	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- A = B   =>    |- (BOUNDED A <-> BOUNDED B )
Theorem	bdceqi 16348	A class equal to a bounded one is bounded. Note the use of ax-ext 2211. See also bdceqir 16349. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- A = B   =>    |- BOUNDED B
Theorem	bdceqir 16349	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16348) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16330). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- B = A   =>    |- BOUNDED B
Theorem	bdel 16350*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- (BOUNDED A -> BOUNDED x e. A )
Theorem	bdeli 16351*	Inference associated with bdel 16350. Its converse is bdelir 16352. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e. A
Theorem	bdelir 16352*	Inference associated with df-bdc 16346. Its converse is bdeli 16351. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. A   =>    |- BOUNDED A
Theorem	bdcv 16353	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x
Theorem	bdcab 16354	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED { x | ph }
Theorem	bdph 16355	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
|- BOUNDED { x | ph }   =>    |- BOUNDED ph
Theorem	bds 16356*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 16327; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 16327. (Contributed by BJ, 19-Nov-2019.)
|- BOUNDED ph   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- BOUNDED ps
Theorem	bdcrab 16357*	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 ph   =>    |- BOUNDED { x e. A | ph }
Theorem	bdne 16358	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x =/= y
Theorem	bdnel 16359*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e/ A
Theorem	bdreu 16360*	Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula A. x e. A ph need not be bounded even if A and ph are. Indeed, _V is bounded by bdcvv 16362, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 16329, if A. x e. _V ph were bounded when ph is bounded, then A. x ph would be bounded as well when ph is bounded, which is not the case. The same remark holds with E. , E! , E*. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED E! x e. y ph
Theorem	bdrmo 16361*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED E* x e. y ph
Theorem	bdcvv 16362	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 16363	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 16364. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdsbcALT 16364	Alternate proof of bdsbc 16363. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdccsb 16365	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 16366	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A \ B )
Theorem	bdcun 16367	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A u. B )
Theorem	bdcin 16368	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A i^i B )
Theorem	bdss 16369	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 C_ A
Theorem	bdcnul 16370	The empty class is bounded. See also bdcnulALT 16371. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED (/)
Theorem	bdcnulALT 16371	Alternate proof of bdcnul 16370. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 16349, or use the corresponding characterizations of its elements followed by bdelir 16352. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED (/)
Theorem	bdeq0 16372	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 16373	Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED (/) e. x
Theorem	bdcpw 16374	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED ~P A
Theorem	bdcsn 16375	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x }
Theorem	bdcpr 16376	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y }
Theorem	bdctp 16377	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y , z }
Theorem	bdsnss 16378*	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 } C_ A
Theorem	bdvsn 16379*	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 16380	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <. x , y >.
Theorem	bdcuni 16381	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U. x
Theorem	bdcint 16382	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^| x
Theorem	bdciun 16383*	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 U_ x e. y A
Theorem	bdciin 16384*	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 e. y A
Theorem	bdcsuc 16385	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 16386*	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 16387	Boundedness of the formula "the successor of the setvar x belongs to the setvar y". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED suc x e. y
Theorem	bdcriota 16388*	A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
|- BOUNDED ph   &    |- E! x e. y ph   =>    |- BOUNDED ( iota_ x e. y ph )
14.2.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 16389*	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 4203. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep1 16390*	Version of ax-bdsep 16389 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep2 16391*	Version of ax-bdsep 16389 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 16390 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnft 16392*	Closed form of bdsepnf 16393. Version of ax-bdsep 16389 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 16390 when sufficient. (Contributed by BJ, 19-Oct-2019.)
|- BOUNDED ph   =>    |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )
Theorem	bdsepnf 16393*	Version of ax-bdsep 16389 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 16394. Use bdsep1 16390 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- F/ b ph   &    |- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnfALT 16394*	Alternate proof of bdsepnf 16393, not using bdsepnft 16392. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ b ph   &    |- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdzfauscl 16395*	Closed form of the version of zfauscl 4205 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
|- BOUNDED ph   =>    |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) )
Theorem	bdbm1.3ii 16396*	Bounded version of bm1.3ii 4206. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- E. x A. y ( ph -> y e. x )   =>    |- E. x A. y ( y e. x <-> ph )
Theorem	bj-axemptylem 16397*	Lemma for bj-axempty 16398 and bj-axempty2 16399. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4211 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 16398*	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 4210. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4211 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 16399*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 16398. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4211 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 16400*	nalset 4215 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x