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Type | Label | Description |
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Statement | ||
Theorem | bdsepnf 13401* | Version of ax-bdsep 13397 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 13402. Use bdsep1 13398 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ Ⅎ𝑏𝜑 & ⊢ BOUNDED 𝜑 ⇒ ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | ||
Theorem | bdsepnfALT 13402* | Alternate proof of bdsepnf 13401, not using bdsepnft 13400. (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 13403* | Closed form of the version of zfauscl 4080 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 13404* | Bounded version of bm1.3ii 4081. (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 13405* | Lemma for bj-axempty 13406 and bj-axempty2 13407. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4086 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | ||
Theorem | bj-axempty 13406* | 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 4085. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4086 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ | ||
Theorem | bj-axempty2 13407* | Axiom of the empty set from bounded separation, alternate version to bj-axempty 13406. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4086 instead. (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Theorem | bj-nalset 13408* | nalset 4090 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | ||
Theorem | bj-vprc 13409 | vprc 4092 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ V ∈ V | ||
Theorem | bj-nvel 13410 | nvel 4093 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ V ∈ 𝐴 | ||
Theorem | bj-vnex 13411 | vnex 4091 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ ¬ ∃𝑥 𝑥 = V | ||
Theorem | bdinex1 13412 | Bounded version of inex1 4094. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∩ 𝐵) ∈ V | ||
Theorem | bdinex2 13413 | Bounded version of inex2 4095. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 & ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∩ 𝐴) ∈ V | ||
Theorem | bdinex1g 13414 | Bounded version of inex1g 4096. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | bdssex 13415 | Bounded version of ssex 4097. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) | ||
Theorem | bdssexi 13416 | Bounded version of ssexi 4098. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ 𝐵 ∈ V & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝐴 ∈ V | ||
Theorem | bdssexg 13417 | Bounded version of ssexg 4099. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | bdssexd 13418 | Bounded version of ssexd 4100. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | bdrabexg 13419* | Bounded version of rabexg 4103. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | bj-inex 13420 | 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 13421 | intexr 4107 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) | ||
Theorem | bj-intnexr 13422 | intnexr 4108 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) | ||
Theorem | bj-zfpair2 13423 | Proof of zfpair2 4165 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | bj-prexg 13424 | Proof of prexg 4166 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | ||
Theorem | bj-snexg 13425 | snexg 4140 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | ||
Theorem | bj-snex 13426 | snex 4141 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ∈ V | ||
Theorem | bj-sels 13427* | If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) | ||
Theorem | bj-axun2 13428* | axun2 4390 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | ||
Theorem | bj-uniex2 13429* | uniex2 4391 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
⊢ ∃𝑦 𝑦 = ∪ 𝑥 | ||
Theorem | bj-uniex 13430 | uniex 4392 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∪ 𝐴 ∈ V | ||
Theorem | bj-uniexg 13431 | uniexg 4394 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | ||
Theorem | bj-unex 13432 | unex 4395 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ V | ||
Theorem | bdunexb 13433 | Bounded version of unexb 4396. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | bj-unexg 13434 | unexg 4397 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | bj-sucexg 13435 | sucexg 4451 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | ||
Theorem | bj-sucex 13436 | sucex 4452 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ suc 𝐴 ∈ V | ||
Axiom | ax-bj-d0cl 13437 | Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ DECID 𝜑 | ||
Theorem | bj-d0clsepcl 13438 | Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
⊢ DECID 𝜑 | ||
Syntax | wind 13439 | Syntax for inductive classes. |
wff Ind 𝐴 | ||
Definition | df-bj-ind 13440* | Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) |
⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | ||
Theorem | bj-indsuc 13441 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) | ||
Theorem | bj-indeq 13442 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵)) | ||
Theorem | bj-bdind 13443 | Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED Ind 𝑥 | ||
Theorem | bj-indint 13444* | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} | ||
Theorem | bj-indind 13445* | If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴 ∩ 𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.) |
⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵)) | ||
Theorem | bj-dfom 13446 | Alternate definition of ω, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.) |
⊢ ω = ∩ {𝑥 ∣ Ind 𝑥} | ||
Theorem | bj-omind 13447 | ω is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind ω | ||
Theorem | bj-omssind 13448 | ω is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) | ||
Theorem | bj-ssom 13449* | A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) | ||
Theorem | bj-om 13450* | A set is equal to ω if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)))) | ||
Theorem | bj-2inf 13451* | Two formulations of the axiom of infinity (see ax-infvn 13454 and bj-omex 13455) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) | ||
The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4547 and peano3 4549 already show this. In this section, we prove bj-peano2 13452 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms. | ||
Theorem | bj-peano2 13452 | Constructive proof of peano2 4548. Temporary note: another possibility is to simply replace sucexg 4451 with bj-sucexg 13435 in the proof of peano2 4548. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) | ||
Theorem | peano5set 13453* | Version of peano5 4551 when ω ∩ 𝐴 is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)) | ||
In the absence of full separation, the axiom of infinity has to be stated more precisely, as the existence of the smallest class containing the empty set and the successor of each of its elements. | ||
In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13454) and deduce that the class ω of natural number ordinals is a set (bj-omex 13455). | ||
Axiom | ax-infvn 13454* | Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4541) from which one then proves, using full separation, that the wanted set exists (omex 4546). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.) |
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) | ||
Theorem | bj-omex 13455 | Proof of omex 4546 from ax-infvn 13454. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
⊢ ω ∈ V | ||
In this section, we give constructive proofs of two versions of Peano's fifth postulate. | ||
Theorem | bdpeano5 13456* | Bounded version of peano5 4551. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | speano5 13457* | Version of peano5 4551 when 𝐴 is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction). We also prove Peano's fourth postulate. Together with the results from the previous sections, this proves from the core axioms of CZF (with infinity) that the set of natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers. | ||
Theorem | findset 13458* | Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4552 for a nonconstructive proof of the general case. See bdfind 13459 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)) | ||
Theorem | bdfind 13459* | Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4552 for a nonconstructive proof of the general case. See findset 13458 for a proof when 𝐴 is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω) | ||
Theorem | bj-bdfindis 13460* | Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4553 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4553, finds2 4554, finds1 4555. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) | ||
Theorem | bj-bdfindisg 13461* | Version of bj-bdfindis 13460 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13460 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜏 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) | ||
Theorem | bj-bdfindes 13462 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13460 for explanations. From this version, it is easy to prove the bounded version of findes 4556. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) | ||
Theorem | bj-nn0suc0 13463* | Constructive proof of a variant of nn0suc 4557. For a constructive proof of nn0suc 4557, see bj-nn0suc 13477. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥)) | ||
Theorem | bj-nntrans 13464 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | bj-nntrans2 13465 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → Tr 𝐴) | ||
Theorem | bj-nnelirr 13466 | A natural number does not belong to itself. Version of elirr 4494 for natural numbers, which does not require ax-setind 4490. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | bj-nnen2lp 13467 |
A version of en2lp 4507 for natural numbers, which does not require
ax-setind 4490.
Note: using this theorem and bj-nnelirr 13466, one can remove dependency on ax-setind 4490 from nntri2 6430 and nndcel 6436; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | ||
Theorem | bj-peano4 13468 | Remove from peano4 4550 dependency on ax-setind 4490. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-omtrans 13469 |
The set ω is transitive. A natural number is
included in
ω. Constructive proof of elnn 4559.
The idea is to use bounded induction with the formula 𝑥 ⊆ ω. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with 𝑥 ⊆ 𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | ||
Theorem | bj-omtrans2 13470 | The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Tr ω | ||
Theorem | bj-nnord 13471 | A natural number is an ordinal class. Constructive proof of nnord 4565. Can also be proved from bj-nnelon 13472 if the latter is proved from bj-omssonALT 13476. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | bj-nnelon 13472 | A natural number is an ordinal. Constructive proof of nnon 4563. Can also be proved from bj-omssonALT 13476. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | bj-omord 13473 | The set ω is an ordinal class. Constructive proof of ordom 4560. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Ord ω | ||
Theorem | bj-omelon 13474 | The set ω is an ordinal. Constructive proof of omelon 4562. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ ω ∈ On | ||
Theorem | bj-omsson 13475 | Constructive proof of omsson 4566. See also bj-omssonALT 13476. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
⊢ ω ⊆ On | ||
Theorem | bj-omssonALT 13476 | Alternate proof of bj-omsson 13475. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ω ⊆ On | ||
Theorem | bj-nn0suc 13477* | Proof of (biconditional form of) nn0suc 4557 from the core axioms of CZF. See also bj-nn0sucALT 13491. As a characterization of the elements of ω, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | ||
In this section, we add the axiom of set induction to the core axioms of CZF. | ||
In this section, we prove some variants of the axiom of set induction. | ||
Theorem | setindft 13478* | Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)) | ||
Theorem | setindf 13479* | Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) | ||
Theorem | setindis 13480* | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑)) ⇒ ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑) | ||
Axiom | ax-bdsetind 13481* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑) | ||
Theorem | bdsetindis 13482* | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑)) ⇒ ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑) | ||
Theorem | bj-inf2vnlem1 13483* | Lemma for bj-inf2vn 13487. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴) | ||
Theorem | bj-inf2vnlem2 13484* | Lemma for bj-inf2vnlem3 13485 and bj-inf2vnlem4 13486. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))) | ||
Theorem | bj-inf2vnlem3 13485* | Lemma for bj-inf2vn 13487. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝑍 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍)) | ||
Theorem | bj-inf2vnlem4 13486* | Lemma for bj-inf2vn2 13488. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍)) | ||
Theorem | bj-inf2vn 13487* | A sufficient condition for ω to be a set. See bj-inf2vn2 13488 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω)) | ||
Theorem | bj-inf2vn2 13488* | A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13487. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω)) | ||
Axiom | ax-inf2 13489* | Another axiom of infinity in a constructive setting (see ax-infvn 13454). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) | ||
Theorem | bj-omex2 13490 | Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13454 (see bj-2inf 13451 for the equivalence of the latter with bj-omex 13455). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ω ∈ V | ||
Theorem | bj-nn0sucALT 13491* | Alternate proof of bj-nn0suc 13477, also constructive but from ax-inf2 13489, hence requiring ax-bdsetind 13481. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | ||
In this section, using the axiom of set induction, we prove full induction on the set of natural numbers. | ||
Theorem | bj-findis 13492* | Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 13460 for a bounded version not requiring ax-setind 4490. See finds 4553 for a proof in IZF. From this version, it is easy to prove of finds 4553, finds2 4554, finds1 4555. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) | ||
Theorem | bj-findisg 13493* | Version of bj-findis 13492 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13492 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜏 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) | ||
Theorem | bj-findes 13494 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13492 for explanations. From this version, it is easy to prove findes 4556. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) | ||
In this section, we state the axiom scheme of strong collection, which is part of CZF set theory. | ||
Axiom | ax-strcoll 13495* | Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function on 𝑎, or equivalently a collection of nonempty classes indexed by 𝑎, and the axiom asserts the existence of a set 𝑏 which "collects" at least one element in the image of each 𝑥 ∈ 𝑎 and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4075. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcoll2 13496* | Version of ax-strcoll 13495 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcollnft 13497* | Closed form of strcollnf 13498. (Contributed by BJ, 21-Oct-2019.) |
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) | ||
Theorem | strcollnf 13498* |
Version of ax-strcoll 13495 with one disjoint variable condition
removed,
the other disjoint variable condition replaced with a non-freeness
hypothesis, and without initial universal quantifier. Version of
strcoll2 13496 with the disjoint variable condition on
𝑏, 𝜑 replaced
with a non-freeness hypothesis.
This proof aims to demonstrate a standard technique, but strcoll2 13496 will generally suffice: since the theorem asserts the existence of a set 𝑏, supposing that that setvar does not occur in the already defined 𝜑 is not a big constraint. (Contributed by BJ, 21-Oct-2019.) |
⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcollnfALT 13499* | Alternate proof of strcollnf 13498, not using strcollnft 13497. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
In this section, we state the axiom scheme of subset collection, which is part of CZF set theory. | ||
Axiom | ax-sscoll 13500* | Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function from 𝑎 to 𝑏, or equivalently a collection of nonempty subsets of 𝑏 indexed by 𝑎, and the consequent asserts the existence of a subset of 𝑐 which "collects" at least one element in the image of each 𝑥 ∈ 𝑎 and which is made only of such elements. The axiom asserts the existence, for any sets 𝑎, 𝑏, of a set 𝑐 such that that implication holds for any value of the parameter 𝑧 of 𝜑. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∀𝑎∀𝑏∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)) |
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