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Type | Label | Description |
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Statement | ||
Theorem | dcdc 9901 | 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 9902* | Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1489 and 19.9ht 1532 or 19.23ht 1386). (Proof modification is discouraged.) |
⊢ (∃𝑥𝜑 → 𝜑) | ||
Theorem | bj-hbalt 9903 | Closed form of hbal 1366 (copied from set.mm). (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | ||
Theorem | bj-nfalt 9904 | Closed form of nfal 1468 (copied from set.mm). (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) | ||
Theorem | spimd 9905 | Deduction form of spim 1626. (Contributed by BJ, 17-Oct-2019.) |
⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | 2spim 9906* | Double substitution, as in spim 1626. (Contributed by BJ, 17-Oct-2019.) |
⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑧𝜒 & ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒)) ⇒ ⊢ (∀𝑧∀𝑥𝜓 → 𝜒) | ||
Theorem | ch2var 9907* | 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 9908* | Version of ch2var 9907 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 9909 | Lemma for bj-vtoclgf 9915. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜒 → 𝜑) ⇒ ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓)) | ||
Theorem | bj-exlimmpi 9910 | Lemma for bj-vtoclgf 9915. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜒 → 𝜑) & ⊢ (𝜒 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜒 → 𝜓) | ||
Theorem | bj-sbimedh 9911 | A strengthening of sbiedh 1670 (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 9912 | A strengthening of sbieh 1673 (same proof). (Contributed by BJ, 16-Dec-2019.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓) | ||
Theorem | bj-sbime 9913 | A strengthening of sbie 1674 (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 9914 | Weakening two hypotheses of vtoclgf 2612. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓)) | ||
Theorem | bj-vtoclgf 9915 | Weakening two hypotheses of vtoclgf 2612. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → 𝜑) & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓) | ||
Theorem | elabgf0 9916 | Lemma for elabgf 2685. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | ||
Theorem | elabgft1 9917 | One implication of elabgf 2685, in closed form. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) | ||
Theorem | elabgf1 9918 | One implication of elabgf 2685. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elabgf2 9919 | One implication of elabgf 2685. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | elabf1 9920* | One implication of elabf 2686. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elabf2 9921* | One implication of elabf 2686. (Contributed by BJ, 21-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | elab1 9922* | One implication of elab 2687. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) | ||
Theorem | elab2a 9923* | One implication of elab 2687. (Contributed by BJ, 21-Nov-2019.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | elabg2 9924* | One implication of elabg 2688. (Contributed by BJ, 21-Nov-2019.) |
⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | bj-rspgt 9925 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2653 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 9926 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2653 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 9927* | 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 9928 | 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 9929 | 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 9930 | 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 9931 | 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 9933. 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 9933 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 9934 through ax-bdsb 9942) can be written either in closed or inference form. The fact that ax-bd0 9933 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 9941. For a similar method, see bj-omtrans 10081. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 9970 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 9932 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝜑 | ||
Axiom | ax-bd0 9933 | 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 9934 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 → 𝜓) | ||
Axiom | ax-bdan 9935 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∧ 𝜓) | ||
Axiom | ax-bdor 9936 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ∨ 𝜓) | ||
Axiom | ax-bdn 9937 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ¬ 𝜑 | ||
Axiom | ax-bdal 9938* | 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 9939* | 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 9940 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 = 𝑦 | ||
Axiom | ax-bdel 9941 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝑦 | ||
Axiom | ax-bdsb 9942 | 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 9943 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓) | ||
Theorem | bd0 9944 | A formula equivalent to a bounded one is bounded. See also bd0r 9945. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bd0r 9945 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9944) 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 9946 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝜓 ⇒ ⊢ BOUNDED (𝜑 ↔ 𝜓) | ||
Theorem | bdstab 9947 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED STAB 𝜑 | ||
Theorem | bddc 9948 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED DECID 𝜑 | ||
Theorem | bd3or 9949 | 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 9950 | 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 9951 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdtru 9952 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊤ | ||
Theorem | bdfal 9953 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ⊥ | ||
Theorem | bdnth 9954 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdnthALT 9955 | Alternate proof of bdnth 9954 not using bdfal 9953. Then, bdfal 9953 can be proved from this theorem, using fal 1250. The total number of proof steps would be 17 (for bdnthALT 9955) + 3 = 20, which is more than 8 (for bdfal 9953) + 9 (for bdnth 9954) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ BOUNDED 𝜑 | ||
Theorem | bdxor 9956 | 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 9957* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
⊢ BOUNDED 𝑦 = 𝐴 ⇒ ⊢ BOUNDED 𝐴 ∈ 𝑥 | ||
Theorem | bdab 9958 | 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 9959 | 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 9961. 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 9995), 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 9960 | Syntax for the predicate BOUNDED. |
wff BOUNDED 𝐴 | ||
Definition | df-bdc 9961* | 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 9962 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵) | ||
Theorem | bdceqi 9963 | A class equal to a bounded one is bounded. Note the use of ax-ext 2022. See also bdceqir 9964. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdceqir 9964 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 9963) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 9945). (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 = 𝐴 ⇒ ⊢ BOUNDED 𝐵 | ||
Theorem | bdel 9965* | 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 9966* | Inference associated with bdel 9965. Its converse is bdelir 9967. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝑥 ∈ 𝐴 | ||
Theorem | bdelir 9967* | Inference associated with df-bdc 9961. Its converse is bdeli 9966. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 ∈ 𝐴 ⇒ ⊢ BOUNDED 𝐴 | ||
Theorem | bdcv 9968 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝑥 | ||
Theorem | bdcab 9969 | 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 9970 | 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 9971* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 9942; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 9942. (Contributed by BJ, 19-Nov-2019.) |
⊢ BOUNDED 𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ BOUNDED 𝜓 | ||
Theorem | bdcrab 9972* | 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 9973 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝑥 ≠ 𝑦 | ||
Theorem | bdnel 9974* | 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 9975* |
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 9977, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 9944, 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 9976* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑 | ||
Theorem | bdcvv 9977 | 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 9978 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 9979. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdsbcALT 9979 | Alternate proof of bdsbc 9978. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED [𝑦 / 𝑥]𝜑 | ||
Theorem | bdccsb 9980 | 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 9981 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∖ 𝐵) | ||
Theorem | bdcun 9982 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∪ 𝐵) | ||
Theorem | bdcin 9983 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ BOUNDED (𝐴 ∩ 𝐵) | ||
Theorem | bdss 9984 | 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 9985 | The empty class is bounded. See also bdcnulALT 9986. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED ∅ | ||
Theorem | bdcnulALT 9986 | Alternate proof of bdcnul 9985. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 9964, or use the corresponding characterizations of its elements followed by bdelir 9967. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ BOUNDED ∅ | ||
Theorem | bdeq0 9987 | 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 9988 | Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED ∅ ∈ 𝑥 | ||
Theorem | bdcpw 9989 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ BOUNDED 𝒫 𝐴 | ||
Theorem | bdcsn 9990 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥} | ||
Theorem | bdcpr 9991 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦} | ||
Theorem | bdctp 9992 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED {𝑥, 𝑦, 𝑧} | ||
Theorem | bdsnss 9993* | 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 9994* | 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 9995 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
⊢ BOUNDED ⟨𝑥, 𝑦⟩ | ||
Theorem | bdcuni 9996 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
⊢ BOUNDED ∪ 𝑥 | ||
Theorem | bdcint 9997 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED ∩ 𝑥 | ||
Theorem | bdciun 9998* | 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 9999* | 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 10000 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
⊢ BOUNDED suc 𝑥 |
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