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Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpomen 11301 The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.)
(ω × ω) ≈ ω
 
Theoremxpct 11302 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)
 
Theoremunennn 11303 The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)
((𝐴 ≈ ℕ ∧ 𝐵 ≈ ℕ ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ≈ ℕ)
 
Theoremznnen 11304 The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.)
ℤ ≈ ℕ
 
PART 5  GUIDES AND MISCELLANEA
 
5.1  Guides (conventions, explanations, and examples)
 
5.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

  • Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
  • Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
  • Theorems of propositional calculus - [Heyting].
  • Theorems of pure predicate calculus - Metamath Proof Explorer.
  • Theorems of equality and substitution - Metamath Proof Explorer.
  • Axioms of set theory - [Crosilla].
  • Development of set theory - Chapter 10 of [HoTT].
  • Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
  • Theorems about real numbers - [Geuvers].
 
Theoremconventions 11305 Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.
  • Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 11360 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5633, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
  • Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). For this reason, we generally need to prove we are evaluating functions within their domains and avoid the reverse closure theorems of the Metamath Proof Explorer.
  • Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

  • Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
  • iset.mm versus set.mm names

    Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

    As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

    As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

    We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
apapart df-ap 8035 Yes apadd1 8061, apne 8076
seq3, sum3recursive sequence df-seq3 9819 Yes seq3-1 9842, fsum3 10743
iseq , isumrecursive sequence df-iseq 9818 Yes iseq1 9840, fisum 10742

  • Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/forum/#!forum/metamath.
  • (Contributed by Jim Kingdon, 24-Feb-2020.)

    𝜑       𝜑
     
    5.1.2  Definitional examples
     
    Theoremex-or 11306 Example for ax-io 665. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
    (2 = 3 ∨ 4 = 4)
     
    Theoremex-an 11307 Example for ax-ia1 104. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
    (2 = 2 ∧ 3 = 3)
     
    Theorem1kp2ke3k 11308 Example for df-dec 8847, 1000 + 2000 = 3000.

    This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

    This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

    The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

    This proof heavily relies on the decimal constructor df-dec 8847 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

    (Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

    (1000 + 2000) = 3000
     
    Theoremex-fl 11309 Example for df-fl 9642. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
    ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)
     
    Theoremex-ceil 11310 Example for df-ceil 9643. (Contributed by AV, 4-Sep-2021.)
    ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)
     
    Theoremex-exp 11311 Example for df-exp 9920. (Contributed by AV, 4-Sep-2021.)
    ((5↑2) = 25 ∧ (-3↑-2) = (1 / 9))
     
    Theoremex-fac 11312 Example for df-fac 10099. (Contributed by AV, 4-Sep-2021.)
    (!‘5) = 120
     
    Theoremex-bc 11313 Example for df-bc 10121. (Contributed by AV, 4-Sep-2021.)
    (5C3) = 10
     
    Theoremex-dvds 11314 Example for df-dvds 10890: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
    3 ∥ 6
     
    Theoremex-gcd 11315 Example for df-gcd 11032. (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 11316 (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 11317 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 11318 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 11319 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 11320* Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1534 and 19.9ht 1577 or 19.23ht 1431). (Proof modification is discouraged.)
    (∃𝑥𝜑𝜑)
     
    Theorembj-hbalt 11321 Closed form of hbal 1411 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
    (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥𝑦𝜑))
     
    Theorembj-nfalt 11322 Closed form of nfal 1513 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
    (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
     
    Theoremspimd 11323 Deduction form of spim 1673. (Contributed by BJ, 17-Oct-2019.)
    (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓𝜒))
     
    Theorem2spim 11324* Double substitution, as in spim 1673. (Contributed by BJ, 17-Oct-2019.)
    𝑥𝜒    &   𝑧𝜒    &   ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜓𝜒))       (∀𝑧𝑥𝜓𝜒)
     
    Theoremch2var 11325* 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 11326* Version of ch2var 11325 with non-freeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
    ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))    &   𝜑       𝜓
     
    Theorembj-exlimmp 11327 Lemma for bj-vtoclgf 11333. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    𝑥𝜓    &   (𝜒𝜑)       (∀𝑥(𝜒 → (𝜑𝜓)) → (∃𝑥𝜒𝜓))
     
    Theorembj-exlimmpi 11328 Lemma for bj-vtoclgf 11333. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    𝑥𝜓    &   (𝜒𝜑)    &   (𝜒 → (𝜑𝜓))       (∃𝑥𝜒𝜓)
     
    Theorembj-sbimedh 11329 A strengthening of sbiedh 1717 (same proof). (Contributed by BJ, 16-Dec-2019.)
    (𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
     
    Theorembj-sbimeh 11330 A strengthening of sbieh 1720 (same proof). (Contributed by BJ, 16-Dec-2019.)
    (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
     
    Theorembj-sbime 11331 A strengthening of sbie 1721 (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 11332 Weakening two hypotheses of vtoclgf 2677. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴𝜑)       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉𝜓))
     
    Theorembj-vtoclgf 11333 Weakening two hypotheses of vtoclgf 2677. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴𝜑)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉𝜓)
     
    Theoremelabgf0 11334 Lemma for elabgf 2756. (Contributed by BJ, 21-Nov-2019.)
    (𝑥 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ 𝜑))
     
    Theoremelabgft1 11335 One implication of elabgf 2756, in closed form. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝐴    &   𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
     
    Theoremelabgf1 11336 One implication of elabgf 2756. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} → 𝜓)
     
    Theoremelabgf2 11337 One implication of elabgf 2756. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝐵 → (𝜓𝐴 ∈ {𝑥𝜑}))
     
    Theoremelabf1 11338* One implication of elabf 2757. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} → 𝜓)
     
    Theoremelabf2 11339* One implication of elabf 2757. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝜓𝐴 ∈ {𝑥𝜑})
     
    Theoremelab1 11340* One implication of elab 2758. (Contributed by BJ, 21-Nov-2019.)
    (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ {𝑥𝜑} → 𝜓)
     
    Theoremelab2a 11341* One implication of elab 2758. (Contributed by BJ, 21-Nov-2019.)
    𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝜓𝐴 ∈ {𝑥𝜑})
     
    Theoremelabg2 11342* One implication of elabg 2759. (Contributed by BJ, 21-Nov-2019.)
    (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
     
    Theorembj-rspgt 11343 Restricted specialization, generalized. Weakens a hypothesis of rspccv 2719 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 11344 Restricted specialization, generalized. Weakens a hypothesis of rspccv 2719 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
    𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))
     
    Theoremcbvrald 11345* 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 11346 Version of intss1 3698 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
    𝑥𝐴       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 {𝑥𝜑} ⊆ 𝐴))
     
    Theorembj-intabssel1 11347 Version of intss1 3698 using a class abstraction and implicit substitution. Closed form of intmin3 3710. (Contributed by BJ, 29-Nov-2019.)
    𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓 {𝑥𝜑} ⊆ 𝐴))
     
    Theorembj-elssuniab 11348 Version of elssuni 3676 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
    𝑥𝐴       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 {𝑥𝜑}))
     
    Theorembj-sseq 11349 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  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 𝑥𝐵DECID 𝑥𝐴 (see df-dcin 11351).

    Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 11389).

     
    Syntaxwdcin 11350 Syntax for decidability of a class in another.
    wff 𝐴 DECIDin 𝐵
     
    Definitiondf-dcin 11351* Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
    (𝐴 DECIDin 𝐵 ↔ ∀𝑥𝐵 DECID 𝑥𝐴)
     
    Theoremdecidi 11352 Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
    (𝐴 DECIDin 𝐵 → (𝑋𝐵 → (𝑋𝐴 ∨ ¬ 𝑋𝐴)))
     
    Theoremdecidr 11353* Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
    (𝜑 → (𝑥𝐵 → (𝑥𝐴 ∨ ¬ 𝑥𝐴)))       (𝜑𝐴 DECIDin 𝐵)
     
    Theoremdecidin 11354 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 11355 An upperset of integers is decidable in the integers. Reformulation of eluzdc 9066. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
    (𝑀 ∈ ℤ → (ℤ𝑀) DECIDin ℤ)
     
    Theoremsumdc2 11356* Alternate proof of sumdc 10711, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 10711). (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 11357* Lemma for djulcl 6722 and djurcl 6723. (Contributed by BJ, 4-Jul-2022.)
    𝑋 ∈ V    &   𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)       (𝐴𝐵 → ((𝐹𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
     
    TheoremdjulclALT 11358 Shortening of djulcl 6722 using djucllem 11357. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
    (𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
     
    TheoremdjurclALT 11359 Shortening of djurcl 6723 using djucllem 11357. (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 3949 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 11432. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 3946 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 11534 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 11493. Similarly, the axiom of powerset ax-pow 4001 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 11539.

    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 4343. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 11520.

    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 11361. 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 11361 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 11362 through ax-bdsb 11370) can be written either in closed or inference form. The fact that ax-bd0 11361 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 11369. For a similar method, see bj-omtrans 11508.

    Note that one cannot add an axiom BOUNDED 𝑥𝐴 since by bdph 11398 it would imply that every formula is bounded.

     
    Syntaxwbd 11360 Syntax for the predicate BOUNDED.
    wff BOUNDED 𝜑
     
    Axiomax-bd0 11361 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 11362 An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
     
    Axiomax-bdan 11363 The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
     
    Axiomax-bdor 11364 The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
     
    Axiomax-bdn 11365 The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
    BOUNDED 𝜑       BOUNDED ¬ 𝜑
     
    Axiomax-bdal 11366* 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       BOUNDED𝑥𝑦 𝜑
     
    Axiomax-bdex 11367* 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       BOUNDED𝑥𝑦 𝜑
     
    Axiomax-bdeq 11368 An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝑥 = 𝑦
     
    Axiomax-bdel 11369 An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝑥𝑦
     
    Axiomax-bdsb 11370 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 1693, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑       BOUNDED [𝑦 / 𝑥]𝜑
     
    Theorembdeq 11371 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
    (𝜑𝜓)       (BOUNDED 𝜑BOUNDED 𝜓)
     
    Theorembd0 11372 A formula equivalent to a bounded one is bounded. See also bd0r 11373. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑    &   (𝜑𝜓)       BOUNDED 𝜓
     
    Theorembd0r 11373 A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 11372) 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 11374 A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
     
    Theorembdstab 11375 Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑       BOUNDED STAB 𝜑
     
    Theorembddc 11376 Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑       BOUNDED DECID 𝜑
     
    Theorembd3or 11377 A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓    &   BOUNDED 𝜒       BOUNDED (𝜑𝜓𝜒)
     
    Theorembd3an 11378 A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓    &   BOUNDED 𝜒       BOUNDED (𝜑𝜓𝜒)
     
    Theorembdth 11379 A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
    𝜑       BOUNDED 𝜑
     
    Theorembdtru 11380 The truth value is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED
     
    Theorembdfal 11381 The truth value is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED
     
    Theorembdnth 11382 A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
    ¬ 𝜑       BOUNDED 𝜑
     
    TheorembdnthALT 11383 Alternate proof of bdnth 11382 not using bdfal 11381. Then, bdfal 11381 can be proved from this theorem, using fal 1296. The total number of proof steps would be 17 (for bdnthALT 11383) + 3 = 20, which is more than 8 (for bdfal 11381) + 9 (for bdnth 11382) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
    ¬ 𝜑       BOUNDED 𝜑
     
    Theorembdxor 11384 The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝜑    &   BOUNDED 𝜓       BOUNDED (𝜑𝜓)
     
    Theorembj-bdcel 11385* Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
    BOUNDED 𝑦 = 𝐴       BOUNDED 𝐴𝑥
     
    Theorembdab 11386 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 11387 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 11389. 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 11423), 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 11388 Syntax for the predicate BOUNDED.
    wff BOUNDED 𝐴
     
    Definitiondf-bdc 11389* 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 11390 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
    𝐴 = 𝐵       (BOUNDED 𝐴BOUNDED 𝐵)
     
    Theorembdceqi 11391 A class equal to a bounded one is bounded. Note the use of ax-ext 2070. See also bdceqir 11392. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝐴    &   𝐴 = 𝐵       BOUNDED 𝐵
     
    Theorembdceqir 11392 A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 11391) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 11373). (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝐴    &   𝐵 = 𝐴       BOUNDED 𝐵
     
    Theorembdel 11393* The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
    (BOUNDED 𝐴BOUNDED 𝑥𝐴)
     
    Theorembdeli 11394* Inference associated with bdel 11393. Its converse is bdelir 11395. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝐴       BOUNDED 𝑥𝐴
     
    Theorembdelir 11395* Inference associated with df-bdc 11389. Its converse is bdeli 11394. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝑥𝐴       BOUNDED 𝐴
     
    Theorembdcv 11396 A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
    BOUNDED 𝑥
     
    Theorembdcab 11397 A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
    BOUNDED 𝜑       BOUNDED {𝑥𝜑}
     
    Theorembdph 11398 A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
    BOUNDED {𝑥𝜑}       BOUNDED 𝜑
     
    Theorembds 11399* Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 11370; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 11370. (Contributed by BJ, 19-Nov-2019.)
    BOUNDED 𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       BOUNDED 𝜓
     
    Theorembdcrab 11400* 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}
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