P. 102 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10101-10124   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-inf2 10101*	Another axiom of infinity in a constructive setting (see ax-infvn 10066). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
Theorem	bj-omex2 10102	Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 10066 (see bj-2inf 10062 for the equivalence of the latter with bj-omex 10067). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	bj-nn0sucALT 10103*	Alternate proof of bj-nn0suc 10089, also constructive but from ax-inf2 10101, hence requiring ax-bdsetind 10093. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
6.3.8.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 10104*	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 10072 for a bounded version not requiring ax-setind 4262. See finds 4323 for a proof in IZF. From this version, it is easy to prove of finds 4323, finds2 4324, finds1 4325. (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 10105*	Version of bj-findis 10104 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10104 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 10106	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 10104 for explanations. From this version, it is easy to prove findes 4326. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
6.3.9  Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 10107*	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 10108*	Version of ax-strcoll 10107 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 10109*	Closed form of strcollnf 10110. Version of ax-strcoll 10107 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 10110*	Version of ax-strcoll 10107 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 10111*	Alternate proof of strcollnf 10110, not using strcollnft 10109. (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))
6.3.10  Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 10112*	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 10113*	Version of ax-sscoll 10112 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))
6.3.11  Real numbers
Axiom	ax-ddkcomp 10114	Axiom of Dedekind completeness for Dedekind real numbers: every nonempty upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 10114 should be used in place of construction specific results. In particular, axcaucvg 6974 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝐵) → 𝑥 ≤ 𝐵)))
6.4  Mathbox for David A. Wheeler
6.4.1  Allsome quantifier 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 10115	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 10116	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 10117	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 10118	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 10119	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 10120	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 10121	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 10122	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 10123	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)
6.5  Mathbox for Jim Kingdon
Theorem	qdencn 10124*	The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 9798 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ 𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)}    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ 𝑄 (abs‘(𝑥 − 𝐴)) < 𝐵)

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