Home | Intuitionistic Logic Explorer Theorem List (p. 100 of 101) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvrald 9901* | 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 9902 | Version of intss1 3630 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 9903 | Version of intss1 3630 using a class abstraction and implicit substitution. Closed form of intmin3 3642. (Contributed by BJ, 29-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) | ||
Theorem | bj-elssuniab 9904 | Version of elssuni 3608 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 9905 | 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 9907. 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 9907 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 9908 through ax-bdsb 9916) can be written either in closed or inference form. The fact that ax-bd0 9907 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 9915. For a similar method, see bj-omtrans 10055. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 9944 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 9906 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝜑 | ||
Axiom | ax-bd0 9907 | 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 9908 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 → 𝜓) | ||
Axiom | ax-bdan 9909 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∧ 𝜓) | ||
Axiom | ax-bdor 9910 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∨ 𝜓) | ||
Axiom | ax-bdn 9911 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ¬ 𝜑 | ||
Axiom | ax-bdal 9912* | 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 9913* | 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 9914 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 = 𝑦 | ||
Axiom | ax-bdel 9915 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝑦 | ||
Axiom | ax-bdsb 9916 | 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 1646, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdeq 9917 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓) | ||
Theorem | bd0 9918 | A formula equivalent to a bounded one is bounded. See also bd0r 9919. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bd0r 9919 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9918) 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 9920 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ↔ 𝜓) | ||
Theorem | bdstab 9921 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED STAB 𝜑 | ||
Theorem | bddc 9922 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED DECID 𝜑 | ||
Theorem | bd3or 9923 | 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 9924 | 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 9925 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdtru 9926 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊤ | ||
Theorem | bdfal 9927 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊥ | ||
Theorem | bdnth 9928 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdnthALT 9929 | Alternate proof of bdnth 9928 not using bdfal 9927. Then, bdfal 9927 can be proved from this theorem, using fal 1250. The total number of proof steps would be 17 (for bdnthALT 9929) + 3 = 20, which is more than 8 (for bdfal 9927) + 9 (for bdnth 9928) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdxor 9930 | 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 9931* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
⊢ BOUNDED 𝑦 = 𝐴 ⇒ ⊢ BOUNDED 𝐴 ∈ 𝑥 | ||
Theorem | bdab 9932 | 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 9933 | 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 9935. 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 9969), 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 9934 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝐴 | ||
Definition | df-bdc 9935* | 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 9936 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) | ||
Theorem | bdceqi 9937 | A class equal to a bounded one is bounded. Note the use of ax-ext 2022. See also bdceqir 9938. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdceqir 9938 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 9937) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 9919). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 = 𝐴 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdel 9939* | 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 9940* | Inference associated with bdel 9939. Its converse is bdelir 9941. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝑥 ∈ 𝐴 | ||
Theorem | bdelir 9941* | Inference associated with df-bdc 9935. Its converse is bdeli 9940. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝐴 ⇒ ⊢ BOUNDED 𝐴 | ||
Theorem | bdcv 9942 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 | ||
Theorem | bdcab 9943 | 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 9944 | 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 9945* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 9916; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 9916. (Contributed by BJ, 19-Nov-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bdcrab 9946* | 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 9947 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝑥 ≠ 𝑦 | ||
Theorem | bdnel 9948* | 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 9949* |
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 9951, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 9918, 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 9950* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 | ||
Theorem | bdcvv 9951 | 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 9952 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 9953. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdsbcALT 9953 | Alternate proof of bdsbc 9952. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdccsb 9954 | 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 9955 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∖ 𝐵) | ||
Theorem | bdcun 9956 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∪ 𝐵) | ||
Theorem | bdcin 9957 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∩ 𝐵) | ||
Theorem | bdss 9958 | 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 9959 | The empty class is bounded. See also bdcnulALT 9960. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ∅ | ||
Theorem | bdcnulALT 9960 | Alternate proof of bdcnul 9959. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 9938, or use the corresponding characterizations of its elements followed by bdelir 9941. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED ∅ | ||
Theorem | bdeq0 9961 | 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 9962 | Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED ∅ ∈ 𝑥 | ||
Theorem | bdcpw 9963 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝒫 𝐴 | ||
Theorem | bdcsn 9964 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥} | ||
Theorem | bdcpr 9965 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦} | ||
Theorem | bdctp 9966 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦, 𝑧} | ||
Theorem | bdsnss 9967* | 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 9968* | 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 9969 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
⊢ BOUNDED ⟨𝑥, 𝑦⟩ | ||
Theorem | bdcuni 9970 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
⊢ BOUNDED ∪ 𝑥 | ||
Theorem | bdcint 9971 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED ∩ 𝑥 | ||
Theorem | bdciun 9972* | 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 9973* | 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 9974 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED suc 𝑥 | ||
Theorem | bdeqsuc 9975* | 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 9976 | 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 9977* | 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 9978* | 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 3875. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsep1 9979* | Version of ax-bdsep 9978 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsep2 9980* | Version of ax-bdsep 9978 with one DV condition removed and without initial universal quantifier. Use bdsep1 9979 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnft 9981* | Closed form of bdsepnf 9982. Version of ax-bdsep 9978 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 9979 when sufficient. (Contributed by BJ, 19-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) | ||
Theorem | bdsepnf 9982* | Version of ax-bdsep 9978 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 9983. Use bdsep1 9979 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ Ⅎ𝑏𝜑 & ⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnfALT 9983* | Alternate proof of bdsepnf 9982, not using bdsepnft 9981. (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 9984* | Closed form of the version of zfauscl 3877 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 9985* | Bounded version of bm1.3ii 3878. (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 9986* | Lemma for bj-axempty 9987 and bj-axempty2 9988. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | ||
Theorem | bj-axempty 9987* | 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 3882. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ | ||
Theorem | bj-axempty2 9988* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 9987. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | bj-nalset 9989* | nalset 3887 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | bj-vprc 9990 | vprc 3888 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ V ∈ V | ||
Theorem | bj-nvel 9991 | nvel 3889 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | bj-vnex 9992 | vnex 3890 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | bdinex1 9993 | Bounded version of inex1 3891. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ V | ||
Theorem | bdinex2 9994 | Bounded version of inex2 3892. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | bdinex1g 9995 | Bounded version of inex1g 3893. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | bdssex 9996 | Bounded version of ssex 3894. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | bdssexi 9997 | Bounded version of ssexi 3895. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | bdssexg 9998 | Bounded version of ssexg 3896. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | bdssexd 9999 | Bounded version of ssexd 3897. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | bdrabexg 10000* | Bounded version of rabexg 3900. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |