P. 139 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13801-13900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fmelpw1o 13801	With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 846, which translate to 1o and (/) respectively by iftrue 3530 and iffalse 3533, giving pwtrufal 13990). As proved in if0ab 13800, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , 1o , (/) ) e. ~P 1o
Theorem	bj-charfun 13802*	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 13803*	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 13804*	Alternate proof of bj-charfundc 13803. It was expected to be much shorter since it uses bj-charfun 13802 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 13805*	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 13806*	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 ) = (/) ) ) )
12.2.8  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 4105 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 13879. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4102 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 13977 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 13936. Similarly, the axiom of powerset ax-pow 4158 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 13982. 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 4519. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 13963. 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 13963) 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 13963 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
12.2.8.1  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 13808. 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 13808 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 13809 through ax-bdsb 13817) can be written either in closed or inference form. The fact that ax-bd0 13808 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 13816. For a similar method, see bj-omtrans 13951. Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 13845 it would imply that every formula is bounded.
Syntax	wbd 13807	Syntax for the predicate BOUNDED.
wff BOUNDED ph
Axiom	ax-bd0 13808	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 13809	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph -> ps )
Axiom	ax-bdan 13810	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )
Axiom	ax-bdor 13811	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )
Axiom	ax-bdn 13812	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED -. ph
Axiom	ax-bdal 13813*	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 13814*	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 13815	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x = y
Axiom	ax-bdel 13816	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. y
Axiom	ax-bdsb 13817	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 1756, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [ y / x ] ph
Theorem	bdeq 13818	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )
Theorem	bd0 13819	A formula equivalent to a bounded one is bounded. See also bd0r 13820. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps
Theorem	bd0r 13820	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 13819) 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 13821	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )
Theorem	bdstab 13822	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED STAB ph
Theorem	bddc 13823	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED DECID ph
Theorem	bd3or 13824	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 13825	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 13826	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- ph   =>    |- BOUNDED ph
Theorem	bdtru 13827	The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED T.
Theorem	bdfal 13828	The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED F.
Theorem	bdnth 13829	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdnthALT 13830	Alternate proof of bdnth 13829 not using bdfal 13828. Then, bdfal 13828 can be proved from this theorem, using fal 1355. The total number of proof steps would be 17 (for bdnthALT 13830) + 3 = 20, which is more than 8 (for bdfal 13828) + 9 (for bdnth 13829) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdxor 13831	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 13832*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
|- BOUNDED y = A   =>    |- BOUNDED A e. x
Theorem	bdab 13833	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 13834	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED CondEq ( x = y -> ph )
12.2.8.2  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 13836. 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 13870), 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 13835	Syntax for the predicate BOUNDED.
wff BOUNDED A
Definition	df-bdc 13836*	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 13837	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- A = B   =>    |- (BOUNDED A <-> BOUNDED B )
Theorem	bdceqi 13838	A class equal to a bounded one is bounded. Note the use of ax-ext 2152. See also bdceqir 13839. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- A = B   =>    |- BOUNDED B
Theorem	bdceqir 13839	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 13838) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 13820). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- B = A   =>    |- BOUNDED B
Theorem	bdel 13840*	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 13841*	Inference associated with bdel 13840. Its converse is bdelir 13842. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e. A
Theorem	bdelir 13842*	Inference associated with df-bdc 13836. Its converse is bdeli 13841. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. A   =>    |- BOUNDED A
Theorem	bdcv 13843	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x
Theorem	bdcab 13844	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 13845	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 13846*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 13817; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 13817. (Contributed by BJ, 19-Nov-2019.)
|- BOUNDED ph   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- BOUNDED ps
Theorem	bdcrab 13847*	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 13848	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x =/= y
Theorem	bdnel 13849*	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 13850*	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 13852, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 13819, 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 13851*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED E* x e. y ph
Theorem	bdcvv 13852	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 13853	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 13854. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdsbcALT 13854	Alternate proof of bdsbc 13853. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdccsb 13855	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 13856	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A \ B )
Theorem	bdcun 13857	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A u. B )
Theorem	bdcin 13858	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 13859	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 13860	The empty class is bounded. See also bdcnulALT 13861. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED (/)
Theorem	bdcnulALT 13861	Alternate proof of bdcnul 13860. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 13839, or use the corresponding characterizations of its elements followed by bdelir 13842. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED (/)
Theorem	bdeq0 13862	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 13863	Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED (/) e. x
Theorem	bdcpw 13864	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED ~P A
Theorem	bdcsn 13865	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x }
Theorem	bdcpr 13866	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y }
Theorem	bdctp 13867	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y , z }
Theorem	bdsnss 13868*	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 13869*	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 13870	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <. x , y >.
Theorem	bdcuni 13871	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U. x
Theorem	bdcint 13872	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^| x
Theorem	bdciun 13873*	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 13874*	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 13875	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 13876*	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 13877	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 13878*	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 )
12.2.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 13879*	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 4105. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep1 13880*	Version of ax-bdsep 13879 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 13881*	Version of ax-bdsep 13879 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13880 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnft 13882*	Closed form of bdsepnf 13883. Version of ax-bdsep 13879 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 13880 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 13883*	Version of ax-bdsep 13879 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13884. Use bdsep1 13880 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 13884*	Alternate proof of bdsepnf 13883, not using bdsepnft 13882. (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 13885*	Closed form of the version of zfauscl 4107 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 13886*	Bounded version of bm1.3ii 4108. (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 13887*	Lemma for bj-axempty 13888 and bj-axempty2 13889. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4113 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 13888*	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 4112. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4113 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 13889*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 13888. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4113 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 13890*	nalset 4117 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x
Theorem	bj-vprc 13891	vprc 4119 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 13892	nvel 4120 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 13893	vnex 4118 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 13894	Bounded version of inex1 4121. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 13895	Bounded version of inex2 4122. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 13896	Bounded version of inex1g 4123. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 13897	Bounded version of ssex 4124. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 13898	Bounded version of ssexi 4125. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	bdssexg 13899	Bounded version of ssexg 4126. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	bdssexd 13900	Bounded version of ssexd 4127. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( ph -> B e. C )   &    |- ( ph -> A C_ B )   &    |- BOUNDED A   =>    |- ( ph -> A e. _V )