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Type | Label | Description |
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Statement | ||
Theorem | bj-inf2vnlem2 9401* | Lemma for bj-inf2vnlem3 9402 and bj-inf2vnlem4 9403. 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 9402* | Lemma for bj-inf2vn 9404. (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 9403* | Lemma for bj-inf2vn2 9405. (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 9404* | A sufficient condition for 𝜔 to be a set. See bj-inf2vn2 9405 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 9405* | A sufficient condition for 𝜔 to be a set; unbounded version of bj-inf2vn 9404. (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 9406* | Another axiom of infinity in a constructive setting (see ax-infvn 9375). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
⊢ ∃𝑎∀x(x ∈ 𝑎 ↔ (x = ∅ ∨ ∃y ∈ 𝑎 x = suc y)) | ||
Theorem | bj-omex2 9407 | Using bounded set induction and the strong axiom of infinity, 𝜔 is a set, that is, we recover ax-infvn 9375 (see bj-2inf 9372 for the equivalence of the latter with bj-omex 9376). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜔 ∈ V | ||
Theorem | bj-nn0sucALT 9408* | Alternate proof of bj-nn0suc 9394, also constructive but from ax-inf2 9406, hence requiring ax-bdsetind 9398. (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 9409* | 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 9381 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 9410* | Version of bj-findis 9409 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 9409 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 9411 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 9409 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 9412* | 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 9413* | Version of ax-strcoll 9412 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 9414* | Closed form of strcollnf 9415. Version of ax-strcoll 9412 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 9415* | Version of ax-strcoll 9412 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 9416* | Alternate proof of strcollnf 9415, not using strcollnft 9414. (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 9417* | 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 9418* | Version of ax-sscoll 9417 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 9419 | 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 9420 | 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 9421 | 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 9422 | 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 9423 | 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 9424 | 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 9425 | 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 9426 | 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 9427 | 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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