P. 171 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 17001-17100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	djurclALT 17001	Shortening of djurcl 7393 using djucllem 16999. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )
14.3.7  Miscellaneous
Theorem	funmptd 17002	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5414, then prove funmptd 17002 from it, and then prove funmpt 5415 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 17003*	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	bj-charfun 17004*	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 17005*	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 17006*	Alternate proof of bj-charfundc 17005. It was expected to be much shorter since it uses bj-charfun 17004 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 17007*	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 17008*	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 ) = (/) ) ) )
14.3.8  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 4249 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 17081. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4246 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 17179 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 17138. Similarly, the axiom of powerset ax-pow 4311 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 17184. 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 4684. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 17165. 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 17165) 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 17165 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
14.3.8.1  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 17010. 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 17010 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 17011 through ax-bdsb 17019) can be written either in closed or inference form. The fact that ax-bd0 17010 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 17018. For a similar method, see bj-omtrans 17153. Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 17047 it would imply that every formula is bounded.
Syntax	wbd 17009	Syntax for the predicate BOUNDED.
wff BOUNDED ph
Axiom	ax-bd0 17010	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 17011	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph -> ps )
Axiom	ax-bdan 17012	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )
Axiom	ax-bdor 17013	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )
Axiom	ax-bdn 17014	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED -. ph
Axiom	ax-bdal 17015*	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 17016*	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 17017	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x = y
Axiom	ax-bdel 17018	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. y
Axiom	ax-bdsb 17019	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 1816, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [ y / x ] ph
Theorem	bdeq 17020	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )
Theorem	bd0 17021	A formula equivalent to a bounded one is bounded. See also bd0r 17022. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps
Theorem	bd0r 17022	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 17021) 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 17023	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )
Theorem	bdstab 17024	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED STAB ph
Theorem	bddc 17025	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED DECID ph
Theorem	bd3or 17026	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 17027	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 17028	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- ph   =>    |- BOUNDED ph
Theorem	bdtru 17029	The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED T.
Theorem	bdfal 17030	The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED F.
Theorem	bdnth 17031	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdnthALT 17032	Alternate proof of bdnth 17031 not using bdfal 17030. Then, bdfal 17030 can be proved from this theorem, using fal 1409. The total number of proof steps would be 17 (for bdnthALT 17032) + 3 = 20, which is more than 8 (for bdfal 17030) + 9 (for bdnth 17031) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdxor 17033	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 17034*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
|- BOUNDED y = A   =>    |- BOUNDED A e. x
Theorem	bdab 17035	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 17036	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED CondEq ( x = y -> ph )
14.3.8.2  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 17038. 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 17072), 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 17037	Syntax for the predicate BOUNDED.
wff BOUNDED A
Definition	df-bdc 17038*	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 17039	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- A = B   =>    |- (BOUNDED A <-> BOUNDED B )
Theorem	bdceqi 17040	A class equal to a bounded one is bounded. Note the use of ax-ext 2220. See also bdceqir 17041. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- A = B   =>    |- BOUNDED B
Theorem	bdceqir 17041	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 17040) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 17022). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- B = A   =>    |- BOUNDED B
Theorem	bdel 17042*	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 17043*	Inference associated with bdel 17042. Its converse is bdelir 17044. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e. A
Theorem	bdelir 17044*	Inference associated with df-bdc 17038. Its converse is bdeli 17043. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. A   =>    |- BOUNDED A
Theorem	bdcv 17045	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x
Theorem	bdcab 17046	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 17047	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 17048*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 17019; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 17019. (Contributed by BJ, 19-Nov-2019.)
|- BOUNDED ph   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- BOUNDED ps
Theorem	bdcrab 17049*	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 17050	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x =/= y
Theorem	bdnel 17051*	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 17052*	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 17054, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 17021, 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 17053*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED E* x e. y ph
Theorem	bdcvv 17054	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 17055	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 17056. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdsbcALT 17056	Alternate proof of bdsbc 17055. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdccsb 17057	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 17058	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A \ B )
Theorem	bdcun 17059	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A u. B )
Theorem	bdcin 17060	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 17061	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 17062	The empty class is bounded. See also bdcnulALT 17063. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED (/)
Theorem	bdcnulALT 17063	Alternate proof of bdcnul 17062. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 17041, or use the corresponding characterizations of its elements followed by bdelir 17044. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED (/)
Theorem	bdeq0 17064	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 17065	Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED (/) e. x
Theorem	bdcpw 17066	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED ~P A
Theorem	bdcsn 17067	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x }
Theorem	bdcpr 17068	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y }
Theorem	bdctp 17069	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y , z }
Theorem	bdsnss 17070*	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 17071*	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 17072	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <. x , y >.
Theorem	bdcuni 17073	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U. x
Theorem	bdcint 17074	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^| x
Theorem	bdciun 17075*	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 17076*	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 17077	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 17078*	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 17079	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 17080*	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 )
14.3.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 17081*	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 4249. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep1 17082*	Version of ax-bdsep 17081 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 17083*	Version of ax-bdsep 17081 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 17082 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnft 17084*	Closed form of bdsepnf 17085. Version of ax-bdsep 17081 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 17082 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 17085*	Version of ax-bdsep 17081 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 17086. Use bdsep1 17082 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 17086*	Alternate proof of bdsepnf 17085, not using bdsepnft 17084. (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	bdsepg 17087*	Version of sepg 4251 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 17088*	Bounded version of bm1.3ii 4254. (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 17089*	Lemma for bj-axempty 17090 and bj-axempty2 17091. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4259 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 17090*	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 4258. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4259 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 17091*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 17090. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4259 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 17092*	nalset 4263 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x
Theorem	bj-vprc 17093	vprc 4265 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 17094	nvel 4266 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 17095	vnex 4264 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 17096	Bounded version of inex1 4267. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 17097	Bounded version of inex2 4268. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 17098	Bounded version of inex1g 4269. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 17099	Bounded version of ssex 4270. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 17100	Bounded version of ssexi 4271. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V