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Type | Label | Description |
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Statement | ||
Various utility theorems using FOL and extensionality. | ||
Theorem | bj-vtoclgft 10301 | Weakening two hypotheses of vtoclgf 2629. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓)) | ||
Theorem | bj-vtoclgf 10302 | Weakening two hypotheses of vtoclgf 2629. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → 𝜑) & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | elabgf0 10303 | Lemma for elabgf 2708. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | ||
Theorem | elabgft1 10304 | One implication of elabgf 2708, in closed form. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) | ||
Theorem | elabgf1 10305 | One implication of elabgf 2708. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elabgf2 10306 | One implication of elabgf 2708. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | elabf1 10307* | One implication of elabf 2709. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elabf2 10308* | One implication of elabf 2709. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | elab1 10309* | One implication of elab 2710. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elab2a 10310* | One implication of elab 2710. (Contributed by BJ, 21-Nov-2019.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | elabg2 10311* | One implication of elabg 2711. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | bj-rspgt 10312 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2670 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 10313 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2670 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 10314* | 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 10315 | Version of intss1 3658 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 10316 | Version of intss1 3658 using a class abstraction and implicit substitution. Closed form of intmin3 3670. (Contributed by BJ, 29-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) | ||
Theorem | bj-elssuniab 10317 | Version of elssuni 3636 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 10318 | 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 10320. 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 10320 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 10321 through ax-bdsb 10329) can be written either in closed or inference form. The fact that ax-bd0 10320 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 10328. For a similar method, see bj-omtrans 10468. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 10357 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 10319 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝜑 | ||
Axiom | ax-bd0 10320 | 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 10321 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 → 𝜓) | ||
Axiom | ax-bdan 10322 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∧ 𝜓) | ||
Axiom | ax-bdor 10323 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∨ 𝜓) | ||
Axiom | ax-bdn 10324 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ¬ 𝜑 | ||
Axiom | ax-bdal 10325* | 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 10326* | 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 10327 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 = 𝑦 | ||
Axiom | ax-bdel 10328 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝑦 | ||
Axiom | ax-bdsb 10329 | 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 1662, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdeq 10330 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓) | ||
Theorem | bd0 10331 | A formula equivalent to a bounded one is bounded. See also bd0r 10332. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bd0r 10332 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 10331) 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 10333 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ↔ 𝜓) | ||
Theorem | bdstab 10334 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED STAB 𝜑 | ||
Theorem | bddc 10335 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED DECID 𝜑 | ||
Theorem | bd3or 10336 | 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 10337 | 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 10338 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdtru 10339 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊤ | ||
Theorem | bdfal 10340 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊥ | ||
Theorem | bdnth 10341 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdnthALT 10342 | Alternate proof of bdnth 10341 not using bdfal 10340. Then, bdfal 10340 can be proved from this theorem, using fal 1266. The total number of proof steps would be 17 (for bdnthALT 10342) + 3 = 20, which is more than 8 (for bdfal 10340) + 9 (for bdnth 10341) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdxor 10343 | 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 10344* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
⊢ BOUNDED 𝑦 = 𝐴 ⇒ ⊢ BOUNDED 𝐴 ∈ 𝑥 | ||
Theorem | bdab 10345 | 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 10346 | 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 10348. 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 10382), 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 10347 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝐴 | ||
Definition | df-bdc 10348* | 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 10349 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) | ||
Theorem | bdceqi 10350 | A class equal to a bounded one is bounded. Note the use of ax-ext 2038. See also bdceqir 10351. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdceqir 10351 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 10350) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 10332). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 = 𝐴 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdel 10352* | 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 10353* | Inference associated with bdel 10352. Its converse is bdelir 10354. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝑥 ∈ 𝐴 | ||
Theorem | bdelir 10354* | Inference associated with df-bdc 10348. Its converse is bdeli 10353. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝐴 ⇒ ⊢ BOUNDED 𝐴 | ||
Theorem | bdcv 10355 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 | ||
Theorem | bdcab 10356 | 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 10357 | 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 10358* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10329; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10329. (Contributed by BJ, 19-Nov-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bdcrab 10359* | 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 10360 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝑥 ≠ 𝑦 | ||
Theorem | bdnel 10361* | 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 10362* |
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 10364, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 10331, 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 10363* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 | ||
Theorem | bdcvv 10364 | 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 10365 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 10366. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdsbcALT 10366 | Alternate proof of bdsbc 10365. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdccsb 10367 | 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 10368 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∖ 𝐵) | ||
Theorem | bdcun 10369 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∪ 𝐵) | ||
Theorem | bdcin 10370 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∩ 𝐵) | ||
Theorem | bdss 10371 | 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 10372 | The empty class is bounded. See also bdcnulALT 10373. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ∅ | ||
Theorem | bdcnulALT 10373 | Alternate proof of bdcnul 10372. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 10351, or use the corresponding characterizations of its elements followed by bdelir 10354. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED ∅ | ||
Theorem | bdeq0 10374 | Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) |
⊢ BOUNDED 𝑥 = ∅ | ||
Theorem | bj-bd0el 10375 | Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED ∅ ∈ 𝑥 | ||
Theorem | bdcpw 10376 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝒫 𝐴 | ||
Theorem | bdcsn 10377 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥} | ||
Theorem | bdcpr 10378 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦} | ||
Theorem | bdctp 10379 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦, 𝑧} | ||
Theorem | bdsnss 10380* | Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED {𝑥} ⊆ 𝐴 | ||
Theorem | bdvsn 10381* | Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝑥 = {𝑦} | ||
Theorem | bdop 10382 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
⊢ BOUNDED ⟨𝑥, 𝑦⟩ | ||
Theorem | bdcuni 10383 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
⊢ BOUNDED ∪ 𝑥 | ||
Theorem | bdcint 10384 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED ∩ 𝑥 | ||
Theorem | bdciun 10385* | The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴 | ||
Theorem | bdciin 10386* | The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴 | ||
Theorem | bdcsuc 10387 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED suc 𝑥 | ||
Theorem | bdeqsuc 10388* | Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) |
⊢ BOUNDED 𝑥 = suc 𝑦 | ||
Theorem | bj-bdsucel 10389 | Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED suc 𝑥 ∈ 𝑦 | ||
Theorem | bdcriota 10390* | A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.) |
⊢ BOUNDED 𝜑 & ⊢ ∃!𝑥 ∈ 𝑦 𝜑 ⇒ ⊢ BOUNDED (℩𝑥 ∈ 𝑦 𝜑) | ||
In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory. | ||
Axiom | ax-bdsep 10391* | Axiom scheme of bounded (or restricted, or Δ_{0}) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3903. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsep1 10392* | Version of ax-bdsep 10391 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsep2 10393* | Version of ax-bdsep 10391 with one DV condition removed and without initial universal quantifier. Use bdsep1 10392 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnft 10394* | Closed form of bdsepnf 10395. Version of ax-bdsep 10391 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10392 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) | ||
Theorem | bdsepnf 10395* | Version of ax-bdsep 10391 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10396. Use bdsep1 10392 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ Ⅎ𝑏𝜑 & ⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnfALT 10396* | Alternate proof of bdsepnf 10395, not using bdsepnft 10394. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑏𝜑 & ⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdzfauscl 10397* | Closed form of the version of zfauscl 3905 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | ||
Theorem | bdbm1.3ii 10398* | Bounded version of bm1.3ii 3906. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥) ⇒ ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) | ||
Theorem | bj-axemptylem 10399* | Lemma for bj-axempty 10400 and bj-axempty2 10401. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3911 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | ||
Theorem | bj-axempty 10400* | Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3910. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3911 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ |
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