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Theorem List for Intuitionistic Logic Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremex-bc 11101 Example for df-bc 10053. (Contributed by AV, 4-Sep-2021.)
(5C3) = 10
 
Theoremex-dvds 11102 Example for df-dvds 10679: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
3 ∥ 6
 
Theoremex-gcd 11103 Example for df-gcd 10821. (Contributed by AV, 5-Sep-2021.)
(-6 gcd 9) = 3
 
PART 6  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
 
6.1  Mathboxes for user contributions
 
6.1.1  Mathbox guidelines
 
Theoremmathbox 11104 (This theorem is a dummy placeholder for these guidelines. The name 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 set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it.

Guidelines:

1. If at all possible, please use only nullary class constants for new definitions.

2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm.

4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed.

Notes:

1. We may decide to move some theorems to the main part of set.mm for general use.

2. We may make changes to mathboxes to maintain the overall quality of set.mm. Normally we will let you know if a change might impact what you are working on.

3. If you use theorems from another user's mathbox, we don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let us know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

𝜑       𝜑
 
6.2  Mathbox for BJ
 
6.2.1  Propositional calculus
 
Theoremnnexmid 11105 Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.)
¬ ¬ (𝜑 ∨ ¬ 𝜑)
 
Theoremnndc 11106 Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)
¬ ¬ DECID 𝜑
 
Theoremdcdc 11107 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 𝜑DECID 𝜑)
 
6.2.2  Predicate calculus
 
Theorembj-ex 11108* Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1532 and 19.9ht 1575 or 19.23ht 1429). (Proof modification is discouraged.)
(∃𝑥𝜑𝜑)
 
Theorembj-hbalt 11109 Closed form of hbal 1409 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
 
Theorembj-nfalt 11110 Closed form of nfal 1511 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
(∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 
Theoremspimd 11111 Deduction form of spim 1670. (Contributed by BJ, 17-Oct-2019.)
(𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theorem2spim 11112* Double substitution, as in spim 1670. (Contributed by BJ, 17-Oct-2019.)
𝑥𝜒    &   𝑧𝜒    &   ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜓𝜒))       (∀𝑧𝑥𝜓𝜒)
 
Theoremch2var 11113* Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)
𝑥𝜓    &   𝑧𝜓    &   ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremch2varv 11114* Version of ch2var 11113 with non-freeness hypotheses replaced by DV conditions. (Contributed by BJ, 17-Oct-2019.)
((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))    &   𝜑       𝜓
 
Theorembj-exlimmp 11115 Lemma for bj-vtoclgf 11121. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒𝜑)       (∀𝑥(𝜒 → (𝜑𝜓)) → (∃𝑥𝜒𝜓))
 
Theorembj-exlimmpi 11116 Lemma for bj-vtoclgf 11121. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝜒𝜑)    &   (𝜒 → (𝜑𝜓))       (∃𝑥𝜒𝜓)
 
Theorembj-sbimedh 11117 A strengthening of sbiedh 1714 (same proof). (Contributed by BJ, 16-Dec-2019.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
Theorembj-sbimeh 11118 A strengthening of sbieh 1717 (same proof). (Contributed by BJ, 16-Dec-2019.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
Theorembj-sbime 11119 A strengthening of sbie 1718 (same proof). (Contributed by BJ, 16-Dec-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
6.2.3  Extensionality

Various utility theorems using FOL and extensionality.

 
Theorembj-vtoclgft 11120 Weakening two hypotheses of vtoclgf 2671. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴𝜑)       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉𝜓))
 
Theorembj-vtoclgf 11121 Weakening two hypotheses of vtoclgf 2671. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴𝜑)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉𝜓)
 
Theoremelabgf0 11122 Lemma for elabgf 2749. (Contributed by BJ, 21-Nov-2019.)
(𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
 
Theoremelabgft1 11123 One implication of elabgf 2749, in closed form. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
 
Theoremelabgf1 11124 One implication of elabgf 2749. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} → 𝜓)
 
Theoremelabgf2 11125 One implication of elabgf 2749. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))
 
Theoremelabf1 11126* One implication of elabf 2750. (Contributed by BJ, 21-Nov-2019.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} → 𝜓)
 
Theoremelabf2 11127* One implication of elabf 2750. (Contributed by BJ, 21-Nov-2019.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝜓𝐴 ∈ {𝑥𝜑})
 
Theoremelab1 11128* One implication of elab 2751. (Contributed by BJ, 21-Nov-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} → 𝜓)
 
Theoremelab2a 11129* One implication of elab 2751. (Contributed by BJ, 21-Nov-2019.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝜓𝐴 ∈ {𝑥𝜑})
 
Theoremelabg2 11130* One implication of elabg 2752. (Contributed by BJ, 21-Nov-2019.)
(𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
 
Theorembj-rspgt 11131 Restricted specialization, generalized. Weakens a hypothesis of rspccv 2712 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓)))
 
Theorembj-rspg 11132 Restricted specialization, generalized. Weakens a hypothesis of rspccv 2712 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
 
Theoremcbvrald 11133* Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
 
Theorembj-intabssel 11134 Version of intss1 3686 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
𝑥𝐴       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 {𝑥𝜑} ⊆ 𝐴))
 
Theorembj-intabssel1 11135 Version of intss1 3686 using a class abstraction and implicit substitution. Closed form of intmin3 3698. (Contributed by BJ, 29-Nov-2019.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
 
Theorembj-elssuniab 11136 Version of elssuni 3664 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
𝑥𝐴       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 {𝑥𝜑}))
 
Theorembj-sseq 11137 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 → (𝜓𝐴𝐵))    &   (𝜑 → (𝜒𝐵𝐴))       (𝜑 → ((𝜓𝜒) ↔ 𝐴 = 𝐵))
 
6.2.4  Dedidability 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 𝑥𝐵DECID 𝑥𝐴 (see df-dcin 11139).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 11177).

 
Syntaxwdcin 11138 Syntax for decidability of a class in another.
wff 𝐴 DECIDin 𝐵
 
Definitiondf-dcin 11139* Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
(𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
 
Theoremdecidi 11140 Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
(𝐴 DECIDin 𝐵 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
 
Theoremdecidr 11141* Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
(𝜑 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))       (𝜑𝐴 DECIDin 𝐵)
 
Theoremdecidin 11142 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.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 DECIDin 𝐵)    &   (𝜑𝐵 DECIDin 𝐶)       (𝜑𝐴 DECIDin 𝐶)
 
Theoremuzdcinzz 11143 An upperset of integers is decidable in the integers. Reformulation of eluzdc 9029. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
(𝑀 ∈ ℤ → (ℤ𝑀) DECIDin ℤ)
 
Theoremsumdc2 11144* Alternate proof of sumdc 10639, without DV condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 10639). (Contributed by BJ, 19-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑 → ∀𝑥 ∈ (ℤ𝑀)DECID 𝑥𝐴)    &   (𝜑𝑁 ∈ ℤ)       (𝜑DECID 𝑁𝐴)
 
6.2.5  Disjoint union
 
Theoremdjucllem 11145* Lemma for djulcl 6687 and djurcl 6688. (Contributed by BJ, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)       (𝐴𝐵 → ((𝐹𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
 
TheoremdjulclALT 11146 Shortening of djulcl 6687 using djucllem 11145. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
 
TheoremdjurclALT 11147 Shortening of djurcl 6688 using djucllem 11145. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))
 
6.2.6  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 3932 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 11220. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 3929 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 11322 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 11281. Similarly, the axiom of powerset ax-pow 3984 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 11327.

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 4326. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 11308.

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)

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. (available at https://arxiv.org/abs/1808.05204)

I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.

 
6.2.6.1  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 intuitonistic, 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 11149. Indeed, if we posited it in closed form, then we could prove for instance (𝜑BOUNDED 𝜑) and 𝜑BOUNDED 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 11149 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 11150 through ax-bdsb 11158) can be written either in closed or inference form. The fact that ax-bd0 11149 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 𝑥𝑦 is bounded by ax-bdel 11157. For a similar method, see bj-omtrans 11296.

Note that one cannot add an axiom BOUNDED 𝑥𝐴 since by bdph 11186 it would imply that every formula is bounded.

 
Syntaxwbd 11148 Syntax for the predicate BOUNDED.
wff BOUNDED 𝜑
 
Axiomax-bd0 11149 If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
(𝜑𝜓)       (BOUNDED 𝜑BOUNDED 𝜓)
 
Axiomax-bdim 11150 An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
 
Axiomax-bdan 11151 The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
 
Axiomax-bdor 11152 The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
 
Axiomax-bdn 11153 The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
BOUNDED 𝜑       BOUNDED ¬ 𝜑
 
Axiomax-bdal 11154* A bounded universal quantification of a bounded formula is bounded. Note the DV condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
BOUNDED 𝜑       BOUNDED𝑥𝑦 𝜑
 
Axiomax-bdex 11155* A bounded existential quantification of a bounded formula is bounded. Note the DV condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)
BOUNDED 𝜑       BOUNDED𝑥𝑦 𝜑
 
Axiomax-bdeq 11156 An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝑥 = 𝑦
 
Axiomax-bdel 11157 An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝑥𝑦
 
Axiomax-bdsb 11158 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 1690, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑       BOUNDED [𝑦 / 𝑥]𝜑
 
Theorembdeq 11159 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
(𝜑𝜓)       (BOUNDED 𝜑BOUNDED 𝜓)
 
Theorembd0 11160 A formula equivalent to a bounded one is bounded. See also bd0r 11161. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑    &   (𝜑𝜓)       BOUNDED 𝜓
 
Theorembd0r 11161 A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 11160) 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    &   (𝜓𝜑)       BOUNDED 𝜓
 
Theorembdbi 11162 A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
 
Theorembdstab 11163 Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑       BOUNDED STAB 𝜑
 
Theorembddc 11164 Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑       BOUNDED DECID 𝜑
 
Theorembd3or 11165 A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓    &   BOUNDED 𝜒       BOUNDED (𝜑𝜓𝜒)
 
Theorembd3an 11166 A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓    &   BOUNDED 𝜒       BOUNDED (𝜑𝜓𝜒)
 
Theorembdth 11167 A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
𝜑       BOUNDED 𝜑
 
Theorembdtru 11168 The truth value is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED
 
Theorembdfal 11169 The truth value is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED
 
Theorembdnth 11170 A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
¬ 𝜑       BOUNDED 𝜑
 
TheorembdnthALT 11171 Alternate proof of bdnth 11170 not using bdfal 11169. Then, bdfal 11169 can be proved from this theorem, using fal 1294. The total number of proof steps would be 17 (for bdnthALT 11171) + 3 = 20, which is more than 8 (for bdfal 11169) + 9 (for bdnth 11170) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝜑       BOUNDED 𝜑
 
Theorembdxor 11172 The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
 
Theorembj-bdcel 11173* Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
BOUNDED 𝑦 = 𝐴       BOUNDED 𝐴𝑥
 
Theorembdab 11174 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 ∈ {𝑦𝜑}
 
Theorembdcdeq 11175 Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝜑       BOUNDED CondEq(𝑥 = 𝑦𝜑)
 
6.2.6.2  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 11177. 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 11211), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, BOUNDED 𝜑 BOUNDED ⟨{𝑥𝜑}, ({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 𝐴BOUNDED {𝐴}.

 
Syntaxwbdc 11176 Syntax for the predicate BOUNDED.
wff BOUNDED 𝐴
 
Definitiondf-bdc 11177* 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 𝑥𝐴)
 
Theorembdceq 11178 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
𝐴 = 𝐵       (BOUNDED 𝐴BOUNDED 𝐵)
 
Theorembdceqi 11179 A class equal to a bounded one is bounded. Note the use of ax-ext 2067. See also bdceqir 11180. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   𝐴 = 𝐵       BOUNDED 𝐵
 
Theorembdceqir 11180 A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 11179) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 11161). (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   𝐵 = 𝐴       BOUNDED 𝐵
 
Theorembdel 11181* The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
(BOUNDED 𝐴BOUNDED 𝑥𝐴)
 
Theorembdeli 11182* Inference associated with bdel 11181. Its converse is bdelir 11183. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝐴
 
Theorembdelir 11183* Inference associated with df-bdc 11177. Its converse is bdeli 11182. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝑥𝐴       BOUNDED 𝐴
 
Theorembdcv 11184 A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝑥
 
Theorembdcab 11185 A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
BOUNDED 𝜑       BOUNDED {𝑥𝜑}
 
Theorembdph 11186 A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
BOUNDED {𝑥𝜑}       BOUNDED 𝜑
 
Theorembds 11187* Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 11158; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 11158. (Contributed by BJ, 19-Nov-2019.)
BOUNDED 𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       BOUNDED 𝜓
 
Theorembdcrab 11188* 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 {𝑥𝐴𝜑}
 
Theorembdne 11189 Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥𝑦
 
Theorembdnel 11190* Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝐴
 
Theorembdreu 11191* Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula 𝑥𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 11193, and (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 11160, 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 ∃!𝑥𝑦 𝜑
 
Theorembdrmo 11192* Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝜑       BOUNDED ∃*𝑥𝑦 𝜑
 
Theorembdcvv 11193 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
 
Theorembdsbc 11194 A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 11195. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝜑       BOUNDED [𝑦 / 𝑥]𝜑
 
TheorembdsbcALT 11195 Alternate proof of bdsbc 11194. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED 𝜑       BOUNDED [𝑦 / 𝑥]𝜑
 
Theorembdccsb 11196 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 / 𝑥𝐴
 
Theorembdcdif 11197 The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝐵       BOUNDED (𝐴𝐵)
 
Theorembdcun 11198 The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝐵       BOUNDED (𝐴𝐵)
 
Theorembdcin 11199 The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴    &   BOUNDED 𝐵       BOUNDED (𝐴𝐵)
 
Theorembdss 11200 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 𝑥𝐴
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