P. 162 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16101-16200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elabgf1 16101	One implication of elabgf 2945. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elabgf2 16102	One implication of elabgf 2945. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	elabf1 16103*	One implication of elabf 2946. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elabf2 16104*	One implication of elabf 2946. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	elab1 16105*	One implication of elab 2947. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elab2a 16106*	One implication of elab 2947. (Contributed by BJ, 21-Nov-2019.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	elabg2 16107*	One implication of elabg 2949. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	bj-rspgt 16108	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2904 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))
Theorem	bj-rspg 16109	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2904 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))
Theorem	cbvrald 16110*	Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
Theorem	bj-intabssel 16111	Version of intss1 3937 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
Theorem	bj-intabssel1 16112	Version of intss1 3937 using a class abstraction and implicit substitution. Closed form of intmin3 3949. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
Theorem	bj-elssuniab 16113	Version of elssuni 3915 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))
Theorem	bj-sseq 16114	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.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵))    &   ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 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 ∀𝑥 ∈ 𝐵DECID 𝑥 ∈ 𝐴 (see df-dcin 16116). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16162).
Syntax	wdcin 16115	Syntax for decidability of a class in another.
wff 𝐴 DECIDin 𝐵
Definition	df-dcin 16116*	Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
Theorem	decidi 16117	Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
Theorem	decidr 16118*	Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))    ⇒   ⊢ (𝜑 → 𝐴 DECIDin 𝐵)
Theorem	decidin 16119	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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 DECIDin 𝐵)    &   ⊢ (𝜑 → 𝐵 DECIDin 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 DECIDin 𝐶)
Theorem	uzdcinzz 16120	An upperset of integers is decidable in the integers. Reformulation of eluzdc 9801. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) DECIDin ℤ)
Theorem	sumdc2 16121*	Alternate proof of sumdc 11864, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 11864). (Contributed by BJ, 19-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴)
14.2.6  Disjoint union
Theorem	djucllem 16122*	Lemma for djulcl 7214 and djurcl 7215. (Contributed by BJ, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Theorem	djulclALT 16123	Shortening of djulcl 7214 using djucllem 16122. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurclALT 16124	Shortening of djurcl 7215 using djucllem 16122. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
14.2.7  Miscellaneous
Theorem	funmptd 16125	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5354, then prove funmptd 16125 from it, and then prove funmpt 5355 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fnmptd 16126*	The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	if0ab 16127*	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(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3657, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 7428 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bj-charfun 16128*	Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    ⇒   ⊢ (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfundc 16129*	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.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfundcALT 16130*	Alternate proof of bj-charfundc 16129. It was expected to be much shorter since it uses bj-charfun 16128 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfunr 16131*	If a class 𝐴 has a "weak" characteristic function on a class 𝑋, then negated membership in 𝐴 is decidable (in other words, membership in 𝐴 is testable) in 𝑋. The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as ⊢ (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful. The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)
⊢ (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)
Theorem	bj-charfunbi 16132*	In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function. This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
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 4201 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 16205. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4198 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 16303 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 16262. Similarly, the axiom of powerset ax-pow 4257 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 16308. 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 4628. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 16289. 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 16289) 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 16289 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 𝜑 " is a formula meaning that 𝜑 is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.) A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, ∀𝑥⊤ is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to ⊤ which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded. A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 16134. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (𝜑 → BOUNDED 𝜑) and ⊢ (¬ 𝜑 → BOUNDED 𝜑) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.) Having ax-bd0 16134 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 16135 through ax-bdsb 16143) can be written either in closed or inference form. The fact that ax-bd0 16134 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness. Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that 𝑥 ∈ ω is a bounded formula. However, since ω can be defined as "the 𝑦 such that PHI" a proof using the fact that 𝑥 ∈ ω is bounded can be converted to a proof in iset.mm by replacing ω with 𝑦 everywhere and prepending the antecedent PHI, since 𝑥 ∈ 𝑦 is bounded by ax-bdel 16142. For a similar method, see bj-omtrans 16277. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 16171 it would imply that every formula is bounded.
Syntax	wbd 16133	Syntax for the predicate BOUNDED.
wff BOUNDED 𝜑
Axiom	ax-bd0 16134	If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
Axiom	ax-bdim 16135	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 → 𝜓)
Axiom	ax-bdan 16136	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓)
Axiom	ax-bdor 16137	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓)
Axiom	ax-bdn 16138	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ¬ 𝜑
Axiom	ax-bdal 16139*	A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdex 16140*	A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdeq 16141	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 = 𝑦
Axiom	ax-bdel 16142	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝑦
Axiom	ax-bdsb 16143	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 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdeq 16144	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)
Theorem	bd0 16145	A formula equivalent to a bounded one is bounded. See also bd0r 16146. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bd0r 16146	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16145) biconditional in the hypothesis, to work better with definitions (𝜓 is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdbi 16147	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ↔ 𝜓)
Theorem	bdstab 16148	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED STAB 𝜑
Theorem	bddc 16149	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED DECID 𝜑
Theorem	bd3or 16150	A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓 ∨ 𝜒)
Theorem	bd3an 16151	A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒)
Theorem	bdth 16152	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdtru 16153	The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊤
Theorem	bdfal 16154	The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊥
Theorem	bdnth 16155	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdnthALT 16156	Alternate proof of bdnth 16155 not using bdfal 16154. Then, bdfal 16154 can be proved from this theorem, using fal 1402. The total number of proof steps would be 17 (for bdnthALT 16156) + 3 = 20, which is more than 8 (for bdfal 16154) + 9 (for bdnth 16155) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdxor 16157	The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ⊻ 𝜓)
Theorem	bj-bdcel 16158*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
⊢ BOUNDED 𝑦 = 𝐴    ⇒   ⊢ BOUNDED 𝐴 ∈ 𝑥
Theorem	bdab 16159	Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}
Theorem	bdcdeq 16160	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑)
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 16162. 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 16196), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, ⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ⟨{𝑥 ∣ 𝜑}, ({𝑦, suc 𝑧} × ⟨𝑡, ∅⟩)⟩. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like ⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED {𝐴}.
Syntax	wbdc 16161	Syntax for the predicate BOUNDED.
wff BOUNDED 𝐴
Definition	df-bdc 16162*	Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdceq 16163	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
Theorem	bdceqi 16164	A class equal to a bounded one is bounded. Note the use of ax-ext 2211. See also bdceqir 16165. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdceqir 16165	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16164) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16146). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 = 𝐴    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdel 16166*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdeli 16167*	Inference associated with bdel 16166. Its converse is bdelir 16168. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∈ 𝐴
Theorem	bdelir 16168*	Inference associated with df-bdc 16162. Its converse is bdeli 16167. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝐴    ⇒   ⊢ BOUNDED 𝐴
Theorem	bdcv 16169	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥
Theorem	bdcab 16170	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∣ 𝜑}
Theorem	bdph 16171	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED {𝑥 ∣ 𝜑}    ⇒   ⊢ BOUNDED 𝜑
Theorem	bds 16172*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 16143; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 16143. (Contributed by BJ, 19-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdcrab 16173*	A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bdne 16174	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 ≠ 𝑦
Theorem	bdnel 16175*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∉ 𝐴
Theorem	bdreu 16176*	Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 16178, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 16145, if ∀𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then ∀𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
Theorem	bdrmo 16177*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑
Theorem	bdcvv 16178	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 16179	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 16180. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdsbcALT 16180	Alternate proof of bdsbc 16179. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdccsb 16181	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 16182	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∖ 𝐵)
Theorem	bdcun 16183	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∪ 𝐵)
Theorem	bdcin 16184	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∩ 𝐵)
Theorem	bdss 16185	The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ⊆ 𝐴
Theorem	bdcnul 16186	The empty class is bounded. See also bdcnulALT 16187. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ∅
Theorem	bdcnulALT 16187	Alternate proof of bdcnul 16186. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 16165, or use the corresponding characterizations of its elements followed by bdelir 16168. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED ∅
Theorem	bdeq0 16188	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 16189	Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED ∅ ∈ 𝑥
Theorem	bdcpw 16190	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝒫 𝐴
Theorem	bdcsn 16191	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥}
Theorem	bdcpr 16192	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦}
Theorem	bdctp 16193	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦, 𝑧}
Theorem	bdsnss 16194*	Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED {𝑥} ⊆ 𝐴
Theorem	bdvsn 16195*	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 16196	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED ⟨𝑥, 𝑦⟩
Theorem	bdcuni 16197	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
⊢ BOUNDED ∪ 𝑥
Theorem	bdcint 16198	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED ∩ 𝑥
Theorem	bdciun 16199*	The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴
Theorem	bdciin 16200*	The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴