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Statement | ||
Theorem | djurclALT 12701 | Shortening of djurcl 6889 using djucllem 12699. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
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 4006 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 12774. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4003 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 12872 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 12831. Similarly, the axiom of powerset ax-pow 4058 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 12877. 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 4412. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 12858. 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 12858) 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 12858) I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results. | ||
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 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 12703. 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 12703 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 12704 through ax-bdsb 12712) can be written either in closed or inference form. The fact that ax-bd0 12703 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 12711. For a similar method, see bj-omtrans 12846. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 12740 it would imply that every formula is bounded. | ||
Syntax | wbd 12702 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝜑 | ||
Axiom | ax-bd0 12703 | 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 12704 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 → 𝜓) | ||
Axiom | ax-bdan 12705 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∧ 𝜓) | ||
Axiom | ax-bdor 12706 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∨ 𝜓) | ||
Axiom | ax-bdn 12707 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ¬ 𝜑 | ||
Axiom | ax-bdal 12708* | A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑 | ||
Axiom | ax-bdex 12709* | A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑 | ||
Axiom | ax-bdeq 12710 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 = 𝑦 | ||
Axiom | ax-bdel 12711 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝑦 | ||
Axiom | ax-bdsb 12712 | 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 1719, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdeq 12713 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓) | ||
Theorem | bd0 12714 | A formula equivalent to a bounded one is bounded. See also bd0r 12715. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bd0r 12715 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 12714) 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 12716 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ↔ 𝜓) | ||
Theorem | bdstab 12717 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED STAB 𝜑 | ||
Theorem | bddc 12718 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED DECID 𝜑 | ||
Theorem | bd3or 12719 | 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 12720 | 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 12721 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdtru 12722 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊤ | ||
Theorem | bdfal 12723 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊥ | ||
Theorem | bdnth 12724 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdnthALT 12725 | Alternate proof of bdnth 12724 not using bdfal 12723. Then, bdfal 12723 can be proved from this theorem, using fal 1321. The total number of proof steps would be 17 (for bdnthALT 12725) + 3 = 20, which is more than 8 (for bdfal 12723) + 9 (for bdnth 12724) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdxor 12726 | 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 12727* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
⊢ BOUNDED 𝑦 = 𝐴 ⇒ ⊢ BOUNDED 𝐴 ∈ 𝑥 | ||
Theorem | bdab 12728 | 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 12729 | 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 12731. 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 12765), 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 12730 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝐴 | ||
Definition | df-bdc 12731* | 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 12732 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) | ||
Theorem | bdceqi 12733 | A class equal to a bounded one is bounded. Note the use of ax-ext 2097. See also bdceqir 12734. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdceqir 12734 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 12733) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 12715). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 = 𝐴 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdel 12735* | 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 12736* | Inference associated with bdel 12735. Its converse is bdelir 12737. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝑥 ∈ 𝐴 | ||
Theorem | bdelir 12737* | Inference associated with df-bdc 12731. Its converse is bdeli 12736. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝐴 ⇒ ⊢ BOUNDED 𝐴 | ||
Theorem | bdcv 12738 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 | ||
Theorem | bdcab 12739 | 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 12740 | 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 12741* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 12712; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 12712. (Contributed by BJ, 19-Nov-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bdcrab 12742* | 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 12743 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝑥 ≠ 𝑦 | ||
Theorem | bdnel 12744* | 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 12745* |
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 12747, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 12714, 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 12746* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 | ||
Theorem | bdcvv 12747 | 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 12748 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 12749. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdsbcALT 12749 | Alternate proof of bdsbc 12748. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdccsb 12750 | 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 12751 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∖ 𝐵) | ||
Theorem | bdcun 12752 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∪ 𝐵) | ||
Theorem | bdcin 12753 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∩ 𝐵) | ||
Theorem | bdss 12754 | 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 12755 | The empty class is bounded. See also bdcnulALT 12756. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ∅ | ||
Theorem | bdcnulALT 12756 | Alternate proof of bdcnul 12755. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 12734, or use the corresponding characterizations of its elements followed by bdelir 12737. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED ∅ | ||
Theorem | bdeq0 12757 | 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 12758 | Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED ∅ ∈ 𝑥 | ||
Theorem | bdcpw 12759 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝒫 𝐴 | ||
Theorem | bdcsn 12760 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥} | ||
Theorem | bdcpr 12761 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦} | ||
Theorem | bdctp 12762 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦, 𝑧} | ||
Theorem | bdsnss 12763* | 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 12764* | 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 12765 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
⊢ BOUNDED ⟨𝑥, 𝑦⟩ | ||
Theorem | bdcuni 12766 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
⊢ BOUNDED ∪ 𝑥 | ||
Theorem | bdcint 12767 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED ∩ 𝑥 | ||
Theorem | bdciun 12768* | 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 12769* | 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 12770 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED suc 𝑥 | ||
Theorem | bdeqsuc 12771* | 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 12772 | 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 12773* | 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 12774* | 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 4006. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsep1 12775* | Version of ax-bdsep 12774 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsep2 12776* | Version of ax-bdsep 12774 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 12775 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnft 12777* | Closed form of bdsepnf 12778. Version of ax-bdsep 12774 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 12775 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) | ||
Theorem | bdsepnf 12778* | Version of ax-bdsep 12774 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 12779. Use bdsep1 12775 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ Ⅎ𝑏𝜑 & ⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnfALT 12779* | Alternate proof of bdsepnf 12778, not using bdsepnft 12777. (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 12780* | Closed form of the version of zfauscl 4008 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 12781* | Bounded version of bm1.3ii 4009. (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 12782* | Lemma for bj-axempty 12783 and bj-axempty2 12784. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4014 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | ||
Theorem | bj-axempty 12783* | Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4013. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4014 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ | ||
Theorem | bj-axempty2 12784* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 12783. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4014 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | bj-nalset 12785* | nalset 4018 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | bj-vprc 12786 | vprc 4020 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ V ∈ V | ||
Theorem | bj-nvel 12787 | nvel 4021 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | bj-vnex 12788 | vnex 4019 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | bdinex1 12789 | Bounded version of inex1 4022. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ V | ||
Theorem | bdinex2 12790 | Bounded version of inex2 4023. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | bdinex1g 12791 | Bounded version of inex1g 4024. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | bdssex 12792 | Bounded version of ssex 4025. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | bdssexi 12793 | Bounded version of ssexi 4026. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | bdssexg 12794 | Bounded version of ssexg 4027. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | bdssexd 12795 | Bounded version of ssexd 4028. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | bdrabexg 12796* | Bounded version of rabexg 4031. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | bj-inex 12797 | The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | bj-intexr 12798 | intexr 4035 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) | ||
Theorem | bj-intnexr 12799 | intnexr 4036 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) | ||
Theorem | bj-zfpair2 12800 | Proof of zfpair2 4092 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ {𝑥, 𝑦} ∈ V |
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