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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-uniex2 9301* | uniex2 4139 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
⊢ ∃y y = ∪ x | ||
Theorem | bj-uniex 9302 | uniex 4140 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ A ∈ V ⇒ ⊢ ∪ A ∈ V | ||
Theorem | bj-uniexg 9303 | uniexg 4141 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝑉 → ∪ A ∈ V) | ||
Theorem | bj-unex 9304 | unex 4142 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ (A ∪ B) ∈ V | ||
Theorem | bdunexb 9305 | Bounded version of unexb 4143. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED A & ⊢ BOUNDED B ⇒ ⊢ ((A ∈ V ∧ B ∈ V) ↔ (A ∪ B) ∈ V) | ||
Theorem | bj-unexg 9306 | unexg 4144 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (A ∪ B) ∈ V) | ||
Theorem | bj-sucexg 9307 | sucexg 4190 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝑉 → suc A ∈ V) | ||
Theorem | bj-sucex 9308 | sucex 4191 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ A ∈ V ⇒ ⊢ suc A ∈ V | ||
Axiom | ax-bj-d0cl 9309 | Axiom for Δ_{0}-classical logic. (Contributed by BJ, 2-Jan-2020.) |
⊢ BOUNDED φ ⇒ ⊢ DECID φ | ||
Theorem | bj-notbi 9310 | Equivalence property for negation. TODO: minimize all theorems using notbid 591 and notbii 593. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
⊢ ((φ ↔ ψ) → (¬ φ ↔ ¬ ψ)) | ||
Theorem | bj-notbii 9311 | Inference associated with bj-notbi 9310. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (¬ φ ↔ ¬ ψ) | ||
Theorem | bj-notbid 9312 | Deduction form of bj-notbi 9310. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (¬ ψ ↔ ¬ χ)) | ||
Theorem | bj-dcbi 9313 | Equivalence property for DECID. TODO: solve conflict with dcbi 843; minimize dcbii 746 and dcbid 747 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.) |
⊢ ((φ ↔ ψ) → (DECID φ ↔ DECID ψ)) | ||
Theorem | bj-d0clsepcl 9314 | Δ_{0}-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
⊢ DECID φ | ||
Syntax | wind 9315 | Syntax for inductive classes. |
wff Ind A | ||
Definition | df-bj-ind 9316* | 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 9317 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
⊢ (Ind A → (B ∈ A → suc B ∈ A)) | ||
Theorem | bj-indeq 9318 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
⊢ (A = B → (Ind A ↔ Ind B)) | ||
Theorem | bj-bdind 9319 | Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED Ind x | ||
Theorem | bj-indint 9320* | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind ∩ {x ∈ A ∣ Ind x} | ||
Theorem | bj-dfom 9321 | 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 9322 | 𝜔 is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind 𝜔 | ||
Theorem | bj-omssind 9323 | 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 9324* | 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 9325* | 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 9326* | Two formulations of the axiom of infinity (see ax-infvn 9329 and bj-omex 9330) . (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 do not require the axiom of infinity. We give constructive proofs (only the proof of the second postulate has to be modified). We also prove a preliminary version of the fifth Peano postulate. | ||
Theorem | bj-peano2 9327 | Constructive proof of peano2 4261. Temporary note: another possibility is to simply replace sucexg 4190 with bj-sucexg 9307 in the proof of peano2 4261. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝜔 → suc A ∈ 𝜔) | ||
Theorem | peano5set 9328* | Version of peano5 4264 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 9329) and deduce that the class 𝜔 of finite ordinals is a set (bj-omex 9330). | ||
Axiom | ax-infvn 9329* | 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 4254) from which one then proves (omex 4259) using full separation that the wanted set exists. "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.) |
⊢ ∃x(Ind x ∧ ∀y(Ind y → x ⊆ y)) | ||
Theorem | bj-omex 9330 | Proof of omex 4259 from ax-infvn 9329. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝜔 ∈ V | ||
In this section, we give constructive proofs of the remaining two (the fourth and fifth) Peano postulates. More precisely, we prove from the core axioms of CZF that the set of finite ordinals satisfies the Peano postulates and thus provides a model for the set of natural numbers. | ||
Theorem | bdpeano5 9331* | Bounded version of peano5 4264. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED A ⇒ ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A) | ||
Theorem | speano5 9332* | Version of peano5 4264 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). | ||
Theorem | findset 9333* | Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See bdfind 9334 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 9334* | Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See findset 9333 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 9335* | 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 4266 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4266, finds2 4267, finds1 4268. (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 9336* | Version of bj-bdfindis 9335 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9335 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 9337 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 9335 for explanations. From this version, it is easy to prove the bounded version of findes 4269. (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 9338* | Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9348. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∃x ∈ A A = suc x)) | ||
Theorem | bj-nntrans 9339 | 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 9340 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝜔 → Tr A) | ||
Theorem | bj-nnelirr 9341 | A natural number does not belong to itself. Version of elirr 4224 for natural numbers, which does not require ax-setind 4220. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝜔 → ¬ A ∈ A) | ||
Theorem | bj-nnen2lp 9342 |
A version of en2lp 4232 for natural numbers, which does not require
ax-setind 4220.
Note: using this theorem and bj-nnelirr 9341, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; 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 9343 | Remove from peano4 4263 dependency on ax-setind 4220. 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 9344 |
The set 𝜔 is transitive. A natural number is
included in
𝜔.
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 9345 | The set 𝜔 is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Tr 𝜔 | ||
Theorem | bj-omord 9346 | The set 𝜔 is an ordinal. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Ord 𝜔 | ||
Theorem | bj-omelon 9347 | The set 𝜔 is an ordinal. Constructive proof of omelon 4274. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝜔 ∈ On | ||
Theorem | bj-nn0suc 9348* | Proof of (biconditional form of) nn0suc 4270 from the core axioms of CZF. See also bj-nn0sucALT 9362. 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 9349* | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
⊢ (∀xℲyφ → (∀x(∀y ∈ x [y / x]φ → φ) → ∀xφ)) | ||
Theorem | setindf 9350* | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
⊢ Ⅎyφ ⇒ ⊢ (∀x(∀y ∈ x [y / x]φ → φ) → ∀xφ) | ||
Theorem | setindis 9351* | 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 9352* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
⊢ BOUNDED φ ⇒ ⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ) | ||
Theorem | bdsetindis 9353* | 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 9354* | Lemma for bj-inf2vn 9358. 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 9355* | Lemma for bj-inf2vnlem3 9356 and bj-inf2vnlem4 9357. 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 9356* | Lemma for bj-inf2vn 9358. (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 9357* | Lemma for bj-inf2vn2 9359. (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 9358* | A sufficient condition for 𝜔 to be a set. See bj-inf2vn2 9359 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 9359* | A sufficient condition for 𝜔 to be a set; unbounded version of bj-inf2vn 9358. (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 9360* | Another axiom of infinity in a constructive setting (see ax-infvn 9329). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
⊢ ∃𝑎∀x(x ∈ 𝑎 ↔ (x = ∅ ∨ ∃y ∈ 𝑎 x = suc y)) | ||
Theorem | bj-omex2 9361 | Using bounded set induction and the strong axiom of infinity, 𝜔 is a set, that is, we recover ax-infvn 9329 (see bj-2inf 9326 for the equivalence of the latter with bj-omex 9330). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜔 ∈ V | ||
Theorem | bj-nn0sucALT 9362* | Alternate proof of bj-nn0suc 9348, also constructive but from ax-inf2 9360, hence requiring ax-bdsetind 9352. (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 9363* | 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 9335 for a bounded version not requiring ax-setind 4220. See finds 4266 for a proof in IZF. From this version, it is easy to prove of finds 4266, finds2 4267, finds1 4268. (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 9364* | Version of bj-findis 9363 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 9363 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 9365 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 9363 for explanations. From this version, it is easy to prove findes 4269. (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 9366* | Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∀𝑎(∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)) | ||
Theorem | strcoll2 9367* | Version of ax-strcoll 9366 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)) | ||
Theorem | strcollnft 9368* | Closed form of strcollnf 9369. Version of ax-strcoll 9366 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.) |
⊢ (∀x∀yℲ𝑏φ → (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ))) | ||
Theorem | strcollnf 9369* | Version of ax-strcoll 9366 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.) |
⊢ Ⅎ𝑏φ ⇒ ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)) | ||
Theorem | strcollnfALT 9370* | Alternate proof of strcollnf 9369, not using strcollnft 9368. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑏φ ⇒ ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(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 9371* | Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∀𝑎∀𝑏∃𝑐∀z(∀x ∈ 𝑎 ∃y ∈ 𝑏 φ → ∃𝑑 ∈ 𝑐 ∀y(y ∈ 𝑑 ↔ ∃x ∈ 𝑎 φ)) | ||
Theorem | sscoll2 9372* | Version of ax-sscoll 9371 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∃𝑐∀z(∀x ∈ 𝑎 ∃y ∈ 𝑏 φ → ∃𝑑 ∈ 𝑐 ∀y(y ∈ 𝑑 ↔ ∃x ∈ 𝑎 φ)) | ||
These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like ∀xφ → ψ do not imply that φ is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀!x(φ → ψ), and when restricted (applied to a class) we allow ∀!x ∈ Aφ. The first symbol after the setvar variable must always be ∈ if it is the form applied to a class, and since ∈ cannot begin a wff, it is unambiguous. The → looks like it would be a problem because φ or ψ might include implications, but any implication arrow → within any wff must be surrounded by parentheses, so only the implication arrow of ∀! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome. | ||
Syntax | walsi 9373 | Extend wff definition to include "all some" applied to a top-level implication, which means ψ is true whenever φ is true, and there is at least least one x where φ is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
wff ∀!x(φ → ψ) | ||
Syntax | walsc 9374 | Extend wff definition to include "all some" applied to a class, which means φ is true for all x in A, and there is at least one x in A. (Contributed by David A. Wheeler, 20-Oct-2018.) |
wff ∀!x ∈ Aφ | ||
Definition | df-alsi 9375 | Define "all some" applied to a top-level implication, which means ψ is true whenever φ is true and there is at least one x where φ is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (∀!x(φ → ψ) ↔ (∀x(φ → ψ) ∧ ∃xφ)) | ||
Definition | df-alsc 9376 | Define "all some" applied to a class, which means φ is true for all x in A and there is at least one x in A. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (∀!x ∈ Aφ ↔ (∀x ∈ A φ ∧ ∃x x ∈ A)) | ||
Theorem | alsconv 9377 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
⊢ (∀!x(x ∈ A → φ) ↔ ∀!x ∈ Aφ) | ||
Theorem | alsi1d 9378 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (φ → ∀!x(ψ → χ)) ⇒ ⊢ (φ → ∀x(ψ → χ)) | ||
Theorem | alsi2d 9379 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (φ → ∀!x(ψ → χ)) ⇒ ⊢ (φ → ∃xψ) | ||
Theorem | alsc1d 9380 | Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (φ → ∀!x ∈ Aψ) ⇒ ⊢ (φ → ∀x ∈ A ψ) | ||
Theorem | alsc2d 9381 | Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
⊢ (φ → ∀!x ∈ Aψ) ⇒ ⊢ (φ → ∃x x ∈ A) |
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