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Type | Label | Description | ||||||||||||
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Statement | ||||||||||||||
Theorem | absval2 9201 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.) | ||||||||||||
⊢ (A ∈ ℂ → (abs‘A) = (√‘(((ℜ‘A)↑2) + ((ℑ‘A)↑2)))) | ||||||||||||||
Theorem | abs0 9202 | The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ (abs‘0) = 0 | ||||||||||||||
Theorem | absi 9203 | The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.) | ||||||||||||
⊢ (abs‘i) = 1 | ||||||||||||||
Theorem | abs00bd 9204 | If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.) | ||||||||||||
⊢ (φ → A = 0) ⇒ ⊢ (φ → (abs‘A) = 0) | ||||||||||||||
Theorem | absid 9205 | A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (abs‘A) = A) | ||||||||||||||
Theorem | abs1 9206 | The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) | ||||||||||||
⊢ (abs‘1) = 1 | ||||||||||||||
Theorem | absnid 9207 | A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A ≤ 0) → (abs‘A) = -A) | ||||||||||||||
Theorem | absre 9208 | Absolute value of a real number. (Contributed by NM, 17-Mar-2005.) | ||||||||||||
⊢ (A ∈ ℝ → (abs‘A) = (√‘(A↑2))) | ||||||||||||||
Theorem | nn0abscl 9209 | The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) | ||||||||||||
⊢ (A ∈ ℤ → (abs‘A) ∈ ℕ_{0}) | ||||||||||||||
Theorem | zabscl 9210 | The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.) | ||||||||||||
⊢ (A ∈ ℤ → (abs‘A) ∈ ℤ) | ||||||||||||||
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:
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Theorem | conventions 9211 |
Unless there is a reason to diverge, we follow the conventions of
the Metamath Proof Explorer (aka "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.
Label naming conventions Here are a few of the label naming conventions:
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.
(Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ φ ⇒ ⊢ φ | ||||||||||||||
Theorem | mathbox 9212 |
(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 0-ary 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.) | ||||||||||||
⊢ x = x | ||||||||||||||
Theorem | ax1hfs 9213 | Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.) | ||||||||||||
⊢ (φ → (φ ∧ φ)) | ||||||||||||||
Theorem | nnexmid 9214 | 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) | ||||||||||||||
Theorem | nndc 9215 | 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 | ||||||||||||||
Theorem | dcdc 9216 | 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 φ ↔ DECID φ) | ||||||||||||||
Theorem | bj-ex 9217* | Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1486 and 19.9ht 1529 or 19.23ht 1383). (Proof modification is discouraged.) | ||||||||||||
⊢ (∃xφ → φ) | ||||||||||||||
Theorem | bj-hbalt 9218 | Closed form of hbal 1363 (copied from set.mm). (Contributed by BJ, 2-May-2019.) | ||||||||||||
⊢ (∀y(φ → ∀xφ) → (∀yφ → ∀x∀yφ)) | ||||||||||||||
Theorem | bj-nfalt 9219 | Closed form of nfal 1465 (copied from set.mm). (Contributed by BJ, 2-May-2019.) | ||||||||||||
⊢ (∀xℲyφ → Ⅎy∀xφ) | ||||||||||||||
Theorem | spimd 9220 | Deduction form of spim 1623. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ (φ → Ⅎxχ) & ⊢ (φ → ∀x(x = y → (ψ → χ))) ⇒ ⊢ (φ → (∀xψ → χ)) | ||||||||||||||
Theorem | 2spim 9221* | Double substitution, as in spim 1623. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ Ⅎxχ & ⊢ Ⅎzχ & ⊢ ((x = y ∧ z = 𝑡) → (ψ → χ)) ⇒ ⊢ (∀z∀xψ → χ) | ||||||||||||||
Theorem | ch2var 9222* | Implicit substitution of y for x and 𝑡 for z into a theorem. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ Ⅎzψ & ⊢ ((x = y ∧ z = 𝑡) → (φ ↔ ψ)) & ⊢ φ ⇒ ⊢ ψ | ||||||||||||||
Theorem | ch2varv 9223* | Version of ch2var 9222 with non-freeness hypotheses replaced by DV conditions. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ ((x = y ∧ z = 𝑡) → (φ ↔ ψ)) & ⊢ φ ⇒ ⊢ ψ | ||||||||||||||
Theorem | bj-exlimmp 9224 | Lemma for bj-vtoclgf 9230. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (χ → φ) ⇒ ⊢ (∀x(χ → (φ → ψ)) → (∃xχ → ψ)) | ||||||||||||||
Theorem | bj-exlimmpi 9225 | Lemma for bj-vtoclgf 9230. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (χ → φ) & ⊢ (χ → (φ → ψ)) ⇒ ⊢ (∃xχ → ψ) | ||||||||||||||
Theorem | bj-sbimedh 9226 | A strengthening of sbiedh 1667 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ (φ → ∀xφ) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (x = y → (ψ → χ))) ⇒ ⊢ (φ → ([y / x]ψ → χ)) | ||||||||||||||
Theorem | bj-sbimeh 9227 | A strengthening of sbieh 1670 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ ([y / x]φ → ψ) | ||||||||||||||
Theorem | bj-sbime 9228 | A strengthening of sbie 1671 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ ([y / x]φ → ψ) | ||||||||||||||
Various utility theorems using FOL and extensionality. | ||||||||||||||
Theorem | bj-vtoclgft 9229 | Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → φ) ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ 𝑉 → ψ)) | ||||||||||||||
Theorem | bj-vtoclgf 9230 | Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → φ) & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ 𝑉 → ψ) | ||||||||||||||
Theorem | elabgf0 9231 | Lemma for elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (A ∈ {x ∣ φ} ↔ φ)) | ||||||||||||||
Theorem | elabgft1 9232 | One implication of elabgf 2679, in closed form. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ {x ∣ φ} → ψ)) | ||||||||||||||
Theorem | elabgf1 9233 | One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
Theorem | elabgf2 9234 | One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ B → (ψ → A ∈ {x ∣ φ})) | ||||||||||||||
Theorem | elabf1 9235* | One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
Theorem | elabf2 9236* | One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ A ∈ V & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (ψ → A ∈ {x ∣ φ}) | ||||||||||||||
Theorem | elab1 9237* | One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
Theorem | elab2a 9238* | One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ A ∈ V & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (ψ → A ∈ {x ∣ φ}) | ||||||||||||||
Theorem | elabg2 9239* | One implication of elabg 2682. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ 𝑉 → (ψ → A ∈ {x ∣ φ})) | ||||||||||||||
Theorem | bj-rspgt 9240 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 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 9241 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 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 9242* | 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 9243 | Version of intss1 3621 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 9244 | Version of intss1 3621 using a class abstraction and implicit substitution. Closed form of intmin3 3633. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ 𝑉 → (ψ → ∩ {x ∣ φ} ⊆ A)) | ||||||||||||||
Theorem | bj-elssuniab 9245 | Version of elssuni 3599 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 9246 | 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)) | ||||||||||||||
This is an ongoing project to define bounded formulas, following 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 Δ_{0}) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ_{0}) 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 Δ_{0}-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 (ph_{0} ...) and an axiom "$a wff ph_{0} " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph_{0} -> ps_{0} )", 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 9248. 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 9248 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 9249 through ax-bdsb 9257) can be written either in closed or inference form. The fact that ax-bd0 9248 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 9256. For a similar method, see bj-omtrans 9390. Note that one cannot add an axiom ⊢ BOUNDED x ∈ A since by bdph 9285 it would imply that every formula is bounded. For 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) | ||||||||||||||
Syntax | wbd 9247 | Syntax for the predicate BOUNDED. | ||||||||||||
wff BOUNDED φ | ||||||||||||||
Axiom | ax-bd0 9248 | 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 9249 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ → ψ) | ||||||||||||||
Axiom | ax-bdan 9250 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ∧ ψ) | ||||||||||||||
Axiom | ax-bdor 9251 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ∨ ψ) | ||||||||||||||
Axiom | ax-bdn 9252 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED ¬ φ | ||||||||||||||
Axiom | ax-bdal 9253* | 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 ⇒ ⊢ BOUNDED ∀x ∈ y φ | ||||||||||||||
Axiom | ax-bdex 9254* | 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 ⇒ ⊢ BOUNDED ∃x ∈ y φ | ||||||||||||||
Axiom | ax-bdeq 9255 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED x = y | ||||||||||||||
Axiom | ax-bdel 9256 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED x ∈ y | ||||||||||||||
Axiom | ax-bdsb 9257 | 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 1643, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED [y / x]φ | ||||||||||||||
Theorem | bdeq 9258 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ (φ ↔ ψ) ⇒ ⊢ (BOUNDED φ ↔ BOUNDED ψ) | ||||||||||||||
Theorem | bd0 9259 | A formula equivalent to a bounded one is bounded. See also bd0r 9260. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ (φ ↔ ψ) ⇒ ⊢ BOUNDED ψ | ||||||||||||||
Theorem | bd0r 9260 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9259) 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 9261 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ↔ ψ) | ||||||||||||||
Theorem | bdstab 9262 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED STAB φ | ||||||||||||||
Theorem | bddc 9263 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED DECID φ | ||||||||||||||
Theorem | bd3or 9264 | 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 9265 | 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 9266 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) | ||||||||||||
⊢ φ ⇒ ⊢ BOUNDED φ | ||||||||||||||
Theorem | bdtru 9267 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED ⊤ | ||||||||||||||
Theorem | bdfal 9268 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED ⊥ | ||||||||||||||
Theorem | bdnth 9269 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) | ||||||||||||
⊢ ¬ φ ⇒ ⊢ BOUNDED φ | ||||||||||||||
Theorem | bdnthALT 9270 | Alternate proof of bdnth 9269 not using bdfal 9268. Then, bdfal 9268 can be proved from this theorem, using fal 1249. The total number of proof steps would be 17 (for bdnthALT 9270) + 3 = 20, which is more than 8 (for bdfal 9268) + 9 (for bdnth 9269) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||
⊢ ¬ φ ⇒ ⊢ BOUNDED φ | ||||||||||||||
Theorem | bdxor 9271 | 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 9272* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) | ||||||||||||
⊢ BOUNDED y = A ⇒ ⊢ BOUNDED A ∈ x | ||||||||||||||
Theorem | bdab 9273 | 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 9274 | Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED CondEq(x = y → φ) | ||||||||||||||
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 9276. 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 9310), 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 9275 | Syntax for the predicate BOUNDED. | ||||||||||||
wff BOUNDED A | ||||||||||||||
Definition | df-bdc 9276* | 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 9277 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ A = B ⇒ ⊢ (BOUNDED A ↔ BOUNDED B) | ||||||||||||||
Theorem | bdceqi 9278 | A class equal to a bounded one is bounded. Note the use of ax-ext 2019. See also bdceqir 9279. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ A = B ⇒ ⊢ BOUNDED B | ||||||||||||||
Theorem | bdceqir 9279 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 9278) equality in the hypothesis, to work better with definitions (B is the definiendum that one wants to prove bounded; see comment of bd0r 9260). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ B = A ⇒ ⊢ BOUNDED B | ||||||||||||||
Theorem | bdel 9280* | 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 9281* | Inference associated with bdel 9280. Its converse is bdelir 9282. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A ⇒ ⊢ BOUNDED x ∈ A | ||||||||||||||
Theorem | bdelir 9282* | Inference associated with df-bdc 9276. Its converse is bdeli 9281. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED x ∈ A ⇒ ⊢ BOUNDED A | ||||||||||||||
Theorem | bdcv 9283 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED x | ||||||||||||||
Theorem | bdcab 9284 | 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 9285 | 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 9286* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 9257; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 9257. (Contributed by BJ, 19-Nov-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ BOUNDED ψ | ||||||||||||||
Theorem | bdcrab 9287* | 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 9288 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) | ||||||||||||
⊢ BOUNDED x ≠ y | ||||||||||||||
Theorem | bdnel 9289* | 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 9290* |
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 9292, and ⊢ (∀x ∈ Vφ ↔ ∀xφ) (in minimal propositional calculus), so by bd0 9259, 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 9291* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED ∃*x ∈ y φ | ||||||||||||||
Theorem | bdcvv 9292 | The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ_{0}". (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED V | ||||||||||||||
Theorem | bdsbc 9293 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 9294. (Contributed by BJ, 16-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED [y / x]φ | ||||||||||||||
Theorem | bdsbcALT 9294 | Alternate proof of bdsbc 9293. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED [y / x]φ | ||||||||||||||
Theorem | bdccsb 9295 | 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 9296 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ BOUNDED B ⇒ ⊢ BOUNDED (A ∖ B) | ||||||||||||||
Theorem | bdcun 9297 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ BOUNDED B ⇒ ⊢ BOUNDED (A ∪ B) | ||||||||||||||
Theorem | bdcin 9298 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ BOUNDED B ⇒ ⊢ BOUNDED (A ∩ B) | ||||||||||||||
Theorem | bdss 9299 | 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 9300 | The empty class is bounded. See also bdcnulALT 9301. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED ∅ |
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