P. 163 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16201-16300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ex-fac 16201	Example for df-fac 10965. (Contributed by AV, 4-Sep-2021.)
|- ( ! ` 5 ) = ;; 1 2 0
Theorem	ex-bc 16202	Example for df-bc 10987. (Contributed by AV, 4-Sep-2021.)
|- ( 5 _C 3 ) = ; 1 0
Theorem	ex-dvds 16203	Example for df-dvds 12320: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
|- 3 || 6
Theorem	ex-gcd 16204	Example for df-gcd 12496. (Contributed by AV, 5-Sep-2021.)
|- ( -u 6 gcd 9 ) = 3
PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
14.1  Mathboxes for user contributions
14.1.1  Mathbox guidelines
Theorem	mathbox 16205	(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.) A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm. Guidelines: Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details. (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)
|- ph   =>    |- ph
14.2  Mathbox for BJ
14.2.1  Propositional calculus
Theorem	bj-nnsn 16206	As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
|- ( ( ph -> -. ps ) <-> ( -. -. ph -> -. ps ) )
Theorem	bj-nnor 16207	Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)
|- ( -. -. ( ph \/ ps ) <-> ( -. ph -> -. -. ps ) )
Theorem	bj-nnim 16208	The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ( ph -> ps ) -> ( ph -> -. -. ps ) )
Theorem	bj-nnan 16209	The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )
Theorem	bj-nnclavius 16210	Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
|- ( ( -. ph -> ph ) -> -. -. ph )
Theorem	bj-imnimnn 16211	If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 16210 as its last step. (Contributed by BJ, 27-Oct-2024.)
|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps
14.2.1.1  Stable formulas Some of the following theorems, like bj-sttru 16213 or bj-stfal 16215 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.
Theorem	bj-trst 16212	A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
|- ( ph -> STAB ph )
Theorem	bj-sttru 16213	The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB T.
Theorem	bj-fast 16214	A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> STAB ph )
Theorem	bj-stfal 16215	The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB F.
Theorem	bj-nnst 16216	Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 16462 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( -> , -. ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( -> , <-> , -. )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
|- -. -. STAB ph
Theorem	bj-nnbist 16217	If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if ph is a classical tautology, then -. -. ph is an intuitionistic tautology. Therefore, if ph is a classical tautology, then ph is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16230). (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (STAB ph <-> ph ) )
Theorem	bj-stst 16218	Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)
|- (STAB STAB ph <-> STAB ph )
Theorem	bj-stim 16219	A conjunction with a stable consequent is stable. See stabnot 838 for negation , bj-stan 16220 for conjunction , and bj-stal 16222 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
|- (STAB ps -> STAB ( ph -> ps ) )
Theorem	bj-stan 16220	The conjunction of two stable formulas is stable. See bj-stim 16219 for implication, stabnot 838 for negation, and bj-stal 16222 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
|- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )
Theorem	bj-stand 16221	The conjunction of two stable formulas is stable. Deduction form of bj-stan 16220. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16220 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
|- ( ph -> STAB ps )   &    |- ( ph -> STAB ch )   =>    |- ( ph -> STAB ( ps /\ ch ) )
Theorem	bj-stal 16222	The universal quantification of a stable formula is stable. See bj-stim 16219 for implication, stabnot 838 for negation, and bj-stan 16220 for conjunction. (Contributed by BJ, 24-Nov-2023.)
|- ( A. xSTAB ph -> STAB A. x ph )
Theorem	bj-pm2.18st 16223	Clavius law for stable formulas. See pm2.18dc 860. (Contributed by BJ, 4-Dec-2023.)
|- (STAB ph -> ( ( -. ph -> ph ) -> ph ) )
Theorem	bj-con1st 16224	Contraposition when the antecedent is a negated stable proposition. See con1dc 861. (Contributed by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
14.2.1.2  Decidable formulas
Theorem	bj-trdc 16225	A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( ph -> DECID ph )
Theorem	bj-dctru 16226	The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID T.
Theorem	bj-fadc 16227	A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> DECID ph )
Theorem	bj-dcfal 16228	The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID F.
Theorem	bj-dcstab 16229	A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
|- (DECID ph -> STAB ph )
Theorem	bj-nnbidc 16230	If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16217. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (DECID ph <-> ph ) )
Theorem	bj-nndcALT 16231	Alternate proof of nndc 856. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)
|- -. -. DECID ph
Theorem	bj-dcdc 16232	Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)
|- (DECID DECID ph <-> DECID ph )
Theorem	bj-stdc 16233	Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)
|- (STAB DECID ph <-> DECID ph )
Theorem	bj-dcst 16234	Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
|- (DECID STAB ph <-> STAB ph )
14.2.2  Predicate calculus
Theorem	bj-ex 16235*	Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1644 and 19.9ht 1687 or 19.23ht 1543). (Proof modification is discouraged.)
|- ( E. x ph -> ph )
Theorem	bj-hbalt 16236	Closed form of hbal 1523 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
|- ( A. y ( ph -> A. x ph ) -> ( A. y ph -> A. x A. y ph ) )
Theorem	bj-nfalt 16237	Closed form of nfal 1622 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
|- ( A. x F/ y ph -> F/ y A. x ph )
Theorem	spimd 16238	Deduction form of spim 1784. (Contributed by BJ, 17-Oct-2019.)
|- ( ph -> F/ x ch )   &    |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	2spim 16239*	Double substitution, as in spim 1784. (Contributed by BJ, 17-Oct-2019.)
|- F/ x ch   &    |- F/ z ch   &    |- ( ( x = y /\ z = t ) -> ( ps -> ch ) )   =>    |- ( A. z A. x ps -> ch )
Theorem	ch2var 16240*	Implicit substitution of y for x and t for z into a theorem. (Contributed by BJ, 17-Oct-2019.)
|- F/ x ps   &    |- F/ z ps   &    |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	ch2varv 16241*	Version of ch2var 16240 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
|- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	bj-exlimmp 16242	Lemma for bj-vtoclgf 16249. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- ( ch -> ph )   =>    |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )
Theorem	bj-exlimmpi 16243	Lemma for bj-vtoclgf 16249. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- ( ch -> ph )   &    |- ( ch -> ( ph -> ps ) )   =>    |- ( E. x ch -> ps )
Theorem	bj-sbimedh 16244	A strengthening of sbiedh 1833 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( [ y / x ] ps -> ch ) )
Theorem	bj-sbimeh 16245	A strengthening of sbieh 1836 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )
Theorem	bj-sbime 16246	A strengthening of sbie 1837 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )
14.2.3  Set theorey miscellaneous
Theorem	bj-el2oss1o 16247	Shorter proof of el2oss1o 6602 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. 2o -> A C_ 1o )
14.2.4  Extensionality Various utility theorems using FOL and extensionality.
Theorem	bj-vtoclgft 16248	Weakening two hypotheses of vtoclgf 2859. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )
Theorem	bj-vtoclgf 16249	Weakening two hypotheses of vtoclgf 2859. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. V -> ps )
Theorem	elabgf0 16250	Lemma for elabgf 2945. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( A e. { x | ph } <-> ph ) )
Theorem	elabgft1 16251	One implication of elabgf 2945, in closed form. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
Theorem	elabgf1 16252	One implication of elabgf 2945. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elabgf2 16253	One implication of elabgf 2945. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. B -> ( ps -> A e. { x | ph } ) )
Theorem	elabf1 16254*	One implication of elabf 2946. (Contributed by BJ, 21-Nov-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elabf2 16255*	One implication of elabf 2946. (Contributed by BJ, 21-Nov-2019.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )
Theorem	elab1 16256*	One implication of elab 2947. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elab2a 16257*	One implication of elab 2947. (Contributed by BJ, 21-Nov-2019.)
|- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )
Theorem	elabg2 16258*	One implication of elabg 2949. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> A e. { x | ph } ) )
Theorem	bj-rspgt 16259	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2904 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A. x e. B ph -> ( A e. B -> ps ) ) )
Theorem	bj-rspg 16260	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2904 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A. x e. B ph -> ( A e. B -> ps ) )
Theorem	cbvrald 16261*	Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )
Theorem	bj-intabssel 16262	Version of intss1 3938 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )
Theorem	bj-intabssel1 16263	Version of intss1 3938 using a class abstraction and implicit substitution. Closed form of intmin3 3950. (Contributed by BJ, 29-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> |^| { x | ph } C_ A ) )
Theorem	bj-elssuniab 16264	Version of elssuni 3916 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )
Theorem	bj-sseq 16265	If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)
|- ( ph -> ( ps <-> A C_ B ) )   &    |- ( ph -> ( ch <-> B C_ A ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )
14.2.5  Decidability of classes The question of decidability is essential in intuitionistic logic. In intuitionistic set theories, it is natural to define decidability of a set (or class) as decidability of membership in it. One can parameterize this notion with another set (or class) since it is often important to assess decidability of membership in one class among elements of another class. Namely, one will say that " A is decidable in B " if A. x e. BDECID x e. A (see df-dcin 16267). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16313).
Syntax	wdcin 16266	Syntax for decidability of a class in another.
wff A DECIDin B
Definition	df-dcin 16267*	Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
|- ( A DECIDin B <-> A. x e. B DECID x e. A )
Theorem	decidi 16268	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 16269*	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 16270	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 16271	An upperset of integers is decidable in the integers. Reformulation of eluzdc 9822. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
|- ( M e. ZZ -> ( ZZ>= ` M ) DECIDin ZZ )
Theorem	sumdc2 16272*	Alternate proof of sumdc 11890, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11890). (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 16273*	Lemma for djulcl 7234 and djurcl 7235. (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 16274	Shortening of djulcl 7234 using djucllem 16273. (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 16275	Shortening of djurcl 7235 using djucllem 16273. (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 16276	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5358, then prove funmptd 16276 from it, and then prove funmpt 5359 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 16277*	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 16278*	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 3657, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 7448 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
|- if ( ph , A , (/) ) = { x e. A | ph }
Theorem	bj-charfun 16279*	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 16280*	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 16281*	Alternate proof of bj-charfundc 16280. It was expected to be much shorter since it uses bj-charfun 16279 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 16282*	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 16283*	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 4202 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 16356. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4199 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 16454 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 16413. Similarly, the axiom of powerset ax-pow 4259 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 16459. 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 4630. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 16440. 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 16440) 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 16440 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 16285. 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 16285 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 16286 through ax-bdsb 16294) can be written either in closed or inference form. The fact that ax-bd0 16285 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 16293. For a similar method, see bj-omtrans 16428. Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 16322 it would imply that every formula is bounded.
Syntax	wbd 16284	Syntax for the predicate BOUNDED.
wff BOUNDED ph
Axiom	ax-bd0 16285	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 16286	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph -> ps )
Axiom	ax-bdan 16287	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )
Axiom	ax-bdor 16288	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )
Axiom	ax-bdn 16289	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED -. ph
Axiom	ax-bdal 16290*	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 16291*	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 16292	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x = y
Axiom	ax-bdel 16293	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. y
Axiom	ax-bdsb 16294	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 16295	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )
Theorem	bd0 16296	A formula equivalent to a bounded one is bounded. See also bd0r 16297. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps
Theorem	bd0r 16297	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16296) 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 16298	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )
Theorem	bdstab 16299	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED STAB ph
Theorem	bddc 16300	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED DECID ph