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Theorem List for Intuitionistic Logic Explorer - 13601-13694   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstrcollnft 13601* Closed form of strcollnf 13602. (Contributed by BJ, 21-Oct-2019.)
(∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
 
Theoremstrcollnf 13602* Version of ax-strcoll 13599 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 13600 with the disjoint variable condition on 𝑏, 𝜑 replaced with a nonfreeness hypothesis.

This proof aims to demonstrate a standard technique, but strcoll2 13600 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.)

𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 
TheoremstrcollnfALT 13603* Alternate proof of strcollnf 13602, not using strcollnft 13601. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 
11.2.13  CZF: Subset collection

In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.

 
Axiomax-sscoll 13604* 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 𝑥𝑎 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.)
𝑎𝑏𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐 (∀𝑥𝑎𝑦𝑑 𝜑 ∧ ∀𝑦𝑑𝑥𝑎 𝜑))
 
Theoremsscoll2 13605* Version of ax-sscoll 13604 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐 (∀𝑥𝑎𝑦𝑑 𝜑 ∧ ∀𝑦𝑑𝑥𝑎 𝜑))
 
11.2.14  Real numbers
 
Axiomax-ddkcomp 13606 Axiom of Dedekind completeness for Dedekind real numbers: every inhabited 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 Axiom ax-ddkcomp 13606 should be used in place of construction specific results. In particular, axcaucvg 7821 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 𝑦𝑥 ∧ ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵)))
 
11.3  Mathbox for Jim Kingdon
 
11.3.1  Natural numbers
 
Theoremss1oel2o 13607 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4160 which more directly illustrates the contrast with el2oss1o 6391. (Contributed by Jim Kingdon, 8-Aug-2022.)
(EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))
 
Theoremnnti 13608 Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.)
(𝜑𝐴 ∈ ω)       ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢)))
 
Theorem012of 13609 Mapping zero and one between 0 and ω style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐺 ↾ {0, 1}):{0, 1}⟶2o
 
Theorem2o01f 13610 Mapping zero and one between ω and 0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐺 ↾ 2o):2o⟶{0, 1}
 
11.3.2  The power set of a singleton
 
Theorempwtrufal 13611 A subset of the singleton {∅} cannot be anything other than or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4160. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4158), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
(𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))
 
Theorempwle2 13612* An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = 𝑥𝑁 ({𝑥} × 1o)       ((𝑁 ∈ ω ∧ 𝐺:𝑇1-1→𝒫 1o) → 𝑁 ⊆ 2o)
 
Theorempwf1oexmid 13613* An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = 𝑥𝑁 ({𝑥} × 1o)       ((𝑁 ∈ ω ∧ 𝐺:𝑇1-1→𝒫 1o) → (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2oEXMID)))
 
Theoremexmid1stab 13614* If any proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
((𝜑𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})       (𝜑EXMID)
 
Theoremsubctctexmid 13615* If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))    &   (𝜑 → ω ∈ Markov)       (𝜑EXMID)
 
Theoremsssneq 13616* Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
(𝐴 ⊆ {𝐵} → ∀𝑦𝐴𝑧𝐴 𝑦 = 𝑧)
 
Theorempw1nct 13617* A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
(∀𝑟(𝑟 ⊆ (𝒫 1o × ω) → (∀𝑝 ∈ 𝒫 1o𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1o ⊔ 1o))
 
11.3.3  Omniscience of NN+oo
 
Theorem0nninf 13618 The zero element of (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.)
(ω × {∅}) ∈ ℕ
 
Theoremnnsf 13619* Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       𝑆:ℕ⟶ℕ
 
Theorempeano4nninf 13620* The successor function on is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       𝑆:ℕ1-1→ℕ
 
Theorempeano3nninf 13621* The successor function on is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       (𝐴 ∈ ℕ → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
 
Theoremnninfalllem1 13622* Lemma for nninfall 13623. (Contributed by Jim Kingdon, 1-Aug-2022.)
(𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)    &   (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑 → (𝑄𝑃) = ∅)       (𝜑 → ∀𝑛 ∈ ω (𝑃𝑛) = 1o)
 
Theoremnninfall 13623* Given a decidable predicate on , showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which 𝑄 is a decidable predicate is that it assigns a value of either or 1o (which can be thought of as false and true) to every element of . Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
(𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)    &   (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)       (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)
 
Theoremnninfsellemdc 13624* Lemma for nninfself 13627. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
((𝑄 ∈ (2o𝑚) ∧ 𝑁 ∈ ω) → DECID𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
 
Theoremnninfsellemcl 13625* Lemma for nninfself 13627. (Contributed by Jim Kingdon, 8-Aug-2022.)
((𝑄 ∈ (2o𝑚) ∧ 𝑁 ∈ ω) → if(∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
 
Theoremnninfsellemsuc 13626* Lemma for nninfself 13627. (Contributed by Jim Kingdon, 6-Aug-2022.)
((𝑄 ∈ (2o𝑚) ∧ 𝐽 ∈ ω) → if(∀𝑘 ∈ suc suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ⊆ if(∀𝑘 ∈ suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅))
 
Theoremnninfself 13627* Domain and range of the selection function for . (Contributed by Jim Kingdon, 6-Aug-2022.)
𝐸 = (𝑞 ∈ (2o𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))       𝐸:(2o𝑚)⟶ℕ
 
Theoremnninfsellemeq 13628* Lemma for nninfsel 13631. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)    &   (𝜑𝑁 ∈ ω)    &   (𝜑 → ∀𝑘𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)    &   (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅))) = ∅)       (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅)))
 
Theoremnninfsellemqall 13629* Lemma for nninfsel 13631. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)    &   (𝜑𝑁 ∈ ω)       (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅))) = 1o)
 
Theoremnninfsellemeqinf 13630* Lemma for nninfsel 13631. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)       (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ 1o))
 
Theoremnninfsel 13631* 𝐸 is a selection function for . Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.)
𝐸 = (𝑞 ∈ (2o𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))    &   (𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)       (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)
 
Theoremnninfomnilem 13632* Lemma for nninfomni 13633. (Contributed by Jim Kingdon, 10-Aug-2022.)
𝐸 = (𝑞 ∈ (2o𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))        ∈ Omni
 
Theoremnninfomni 13633 is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
∈ Omni
 
Theoremnninffeq 13634* Equality of two functions on which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, (𝜑 → ∀𝑛 ∈ suc ω...). (Contributed by Jim Kingdon, 4-Aug-2023.)
(𝜑𝐹:ℕ⟶ℕ0)    &   (𝜑𝐺:ℕ⟶ℕ0)    &   (𝜑 → (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑥 ∈ ω ↦ 1o)))    &   (𝜑 → ∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))))       (𝜑𝐹 = 𝐺)
 
11.3.4  Schroeder-Bernstein Theorem
 
Theoremexmidsbthrlem 13635* Lemma for exmidsbthr 13636. (Contributed by Jim Kingdon, 11-Aug-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
 
Theoremexmidsbthr 13636* The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
(∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
 
Theoremexmidsbth 13637* The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6912) is the proof of the Schroeder-Bernstein Theorem from the Metamath Proof Explorer database (in which excluded middle holds), but adapted to use EXMID as an antecedent rather than being unconditionally true, as in the non-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 6912.

The reverse direction (exmidsbthr 13636) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)

(EXMID ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
 
Theoremsbthomlem 13638 Lemma for sbthom 13639. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
(𝜑 → ω ∈ Omni)    &   (𝜑𝑌 ⊆ {∅})    &   (𝜑𝐹:ω–1-1-onto→(𝑌 ⊔ ω))       (𝜑 → (𝑌 = ∅ ∨ 𝑌 = {∅}))
 
Theoremsbthom 13639 Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 13637 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ω? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID)
 
11.3.5  Real and complex numbers
 
Theoremqdencn 13640* 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 11106 (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+) → ∃𝑥𝑄 (abs‘(𝑥𝐴)) < 𝐵)
 
Theoremrefeq 13641* Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐺:ℝ⟶ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑥 < 0 → (𝐹𝑥) = (𝐺𝑥)))    &   (𝜑 → ∀𝑥 ∈ ℝ (0 < 𝑥 → (𝐹𝑥) = (𝐺𝑥)))    &   (𝜑 → (𝐹‘0) = (𝐺‘0))       (𝜑𝐹 = 𝐺)
 
Theoremtriap 13642 Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵𝐴 = 𝐵𝐵 < 𝐴) ↔ DECID 𝐴 # 𝐵))
 
Theoremisomninnlem 13643* Lemma for isomninn 13644. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐴𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥𝐴 (𝑓𝑥) = 0 ∨ ∀𝑥𝐴 (𝑓𝑥) = 1)))
 
Theoremisomninn 13644* Omniscience stated in terms of natural numbers. Similar to isomnimap 7081 but it will sometimes be more convenient to use 0 and 1 rather than and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.)
(𝐴𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥𝐴 (𝑓𝑥) = 0 ∨ ∀𝑥𝐴 (𝑓𝑥) = 1)))
 
Theoremcvgcmp2nlemabs 13645* Lemma for cvgcmp2n 13646. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting (seq1( + , 𝐺)‘𝑁) as the sum of (seq1( + , 𝐺)‘𝑀) and a term which gets smaller as 𝑀 gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐺𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ≤ (1 / (2↑𝑘)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))
 
Theoremcvgcmp2n 13646* A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐺𝑘))    &   ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ≤ (1 / (2↑𝑘)))       (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
 
Theoremiooref1o 13647 A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
𝐹 = (𝑥 ∈ ℝ ↦ (1 / (1 + (exp‘𝑥))))       𝐹:ℝ–1-1-onto→(0(,)1)
 
Theoremiooreen 13648 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
(0(,)1) ≈ ℝ
 
11.3.6  Analytic omniscience principles

Omniscience principles refer to several propositions, most of them weaker than full excluded middle, which do not follow from the axioms of IZF set theory.

They are: (0) the Principle of Omniscience (PO), which is another name for excluded middle (see exmidomni 7086), (1) the Limited Principle of Omniscience (LPO) is ω ∈ Omni (see df-omni 7079), (2) the Weak Limited Principle of Omniscience (WLPO) is ω ∈ WOmni (see df-womni 7108), (3) Markov's Principle (MP) is ω ∈ Markov (see df-markov 7096), (4) the Lesser Limited Principle of Omniscience (LLPO) is not yet defined in iset.mm.

They also have analytic counterparts each of which follows from the corresponding omniscience principle: (1) Analytic LPO is real number trichotomy, 𝑥 ∈ ℝ∀𝑦 ∈ ℝ(𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) (see trilpo 13656), (2) Analytic WLPO is decidability of real number equality, 𝑥 ∈ ℝ∀𝑦 ∈ ℝDECID 𝑥 = 𝑦 (see redcwlpo 13668), (3) Analytic MP is 𝑥 ∈ ℝ∀𝑦 ∈ ℝ(𝑥𝑦𝑥 # 𝑦) (see neapmkv 13680), (4) Analytic LLPO is real number dichotomy, 𝑥 ∈ ℝ∀𝑦 ∈ ℝ(𝑥𝑦𝑦𝑥) (most relevant current theorem is maxclpr 11126).

 
Theoremtrilpolemclim 13649* Lemma for trilpo 13656. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐺 = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹𝑛)))       (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
 
Theoremtrilpolemcl 13650* Lemma for trilpo 13656. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))       (𝜑𝐴 ∈ ℝ)
 
Theoremtrilpolemisumle 13651* Lemma for trilpo 13656. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℕ)       (𝜑 → Σ𝑖𝑍 ((1 / (2↑𝑖)) · (𝐹𝑖)) ≤ Σ𝑖𝑍 (1 / (2↑𝑖)))
 
Theoremtrilpolemgt1 13652* Lemma for trilpo 13656. The 1 < 𝐴 case. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))       (𝜑 → ¬ 1 < 𝐴)
 
Theoremtrilpolemeq1 13653* Lemma for trilpo 13656. The 𝐴 = 1 case. This is proved by noting that if any (𝐹𝑥) is zero, then the infinite sum 𝐴 is less than one based on the term which is zero. We are using the fact that the 𝐹 sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))    &   (𝜑𝐴 = 1)       (𝜑 → ∀𝑥 ∈ ℕ (𝐹𝑥) = 1)
 
Theoremtrilpolemlt1 13654* Lemma for trilpo 13656. The 𝐴 < 1 case. We can use the distance between 𝐴 and one (that is, 1 − 𝐴) to find a position in the sequence 𝑛 where terms after that point will not add up to as much as 1 − 𝐴. By finomni 7084 we know the terms up to 𝑛 either contain a zero or are all one. But if they are all one that contradicts the way we constructed 𝑛, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))    &   (𝜑𝐴 < 1)       (𝜑 → ∃𝑥 ∈ ℕ (𝐹𝑥) = 0)
 
Theoremtrilpolemres 13655* Lemma for trilpo 13656. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))    &   (𝜑 → (𝐴 < 1 ∨ 𝐴 = 1 ∨ 1 < 𝐴))       (𝜑 → (∃𝑥 ∈ ℕ (𝐹𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹𝑥) = 1))
 
Theoremtrilpo 13656* Real number trichotomy implies the Limited Principle of Omniscience (LPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 13654 (which means the sequence contains a zero), trilpolemeq1 13653 (which means the sequence is all ones), and trilpolemgt1 13652 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 13642) or that the real numbers are a discrete field (see trirec0 13657).

LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10146 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ω ∈ Omni)
 
Theoremtrirec0 13657* Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy.

This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 13656). (Contributed by Jim Kingdon, 10-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0))
 
Theoremtrirec0xor 13658* Version of trirec0 13657 with exclusive-or.

The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0))
 
Theoremapdifflemf 13659 Lemma for apdiff 13661. Being apart from the point halfway between 𝑄 and 𝑅 suffices for 𝐴 to be a different distance from 𝑄 and from 𝑅. (Contributed by Jim Kingdon, 18-May-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑄 ∈ ℚ)    &   (𝜑𝑅 ∈ ℚ)    &   (𝜑𝑄 < 𝑅)    &   (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)       (𝜑 → (abs‘(𝐴𝑄)) # (abs‘(𝐴𝑅)))
 
Theoremapdifflemr 13660 Lemma for apdiff 13661. (Contributed by Jim Kingdon, 19-May-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑆 ∈ ℚ)    &   (𝜑 → (abs‘(𝐴 − -1)) # (abs‘(𝐴 − 1)))    &   ((𝜑𝑆 ≠ 0) → (abs‘(𝐴 − 0)) # (abs‘(𝐴 − (2 · 𝑆))))       (𝜑𝐴 # 𝑆)
 
Theoremapdiff 13661* The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)
(𝐴 ∈ ℝ → (∀𝑞 ∈ ℚ 𝐴 # 𝑞 ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞𝑟 → (abs‘(𝐴𝑞)) # (abs‘(𝐴𝑟)))))
 
Theoremiswomninnlem 13662* Lemma for iswomnimap 7110. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐴𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID𝑥𝐴 (𝑓𝑥) = 1))
 
Theoremiswomninn 13663* Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7110 but it will sometimes be more convenient to use 0 and 1 rather than and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID𝑥𝐴 (𝑓𝑥) = 1))
 
Theoremiswomni0 13664* Weak omniscience stated in terms of equality with 0. Like iswomninn 13663 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
(𝐴𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID𝑥𝐴 (𝑓𝑥) = 0))
 
Theoremismkvnnlem 13665* Lemma for ismkvnn 13666. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐴𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥𝐴 (𝑓𝑥) = 1 → ∃𝑥𝐴 (𝑓𝑥) = 0)))
 
Theoremismkvnn 13666* The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥𝐴 (𝑓𝑥) = 1 → ∃𝑥𝐴 (𝑓𝑥) = 0)))
 
Theoremredcwlpolemeq1 13667* Lemma for redcwlpo 13668. A biconditionalized version of trilpolemeq1 13653. (Contributed by Jim Kingdon, 21-Jun-2024.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))       (𝜑 → (𝐴 = 1 ↔ ∀𝑥 ∈ ℕ (𝐹𝑥) = 1))
 
Theoremredcwlpo 13668* Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 13667). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10150 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
 
Theoremtridceq 13669* Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 13656 and redcwlpo 13668). Thus, this is an analytic analogue to lpowlpo 7112. (Contributed by Jim Kingdon, 24-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
 
Theoremredc0 13670* Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
 
Theoremreap0 13671* Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℝ DECID 𝑧 # 0)
 
Theoremdceqnconst 13672* Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 13668 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
(∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓𝑥) ≠ 0))
 
Theoremdcapnconst 13673* Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See trilpo 13656 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 13672 and in fact this theorem can be proved using dceqnconst 13672 as shown at dcapnconstALT 13674. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)

(∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓𝑥) ≠ 0))
 
TheoremdcapnconstALT 13674* Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 13673 by means of dceqnconst 13672. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓𝑥) ≠ 0))
 
Theoremnconstwlpolem0 13675* Lemma for nconstwlpo 13678. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(𝜑𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺𝑖))    &   (𝜑 → ∀𝑥 ∈ ℕ (𝐺𝑥) = 0)       (𝜑𝐴 = 0)
 
Theoremnconstwlpolemgt0 13676* Lemma for nconstwlpo 13678. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
(𝜑𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺𝑖))    &   (𝜑 → ∃𝑥 ∈ ℕ (𝐺𝑥) = 1)       (𝜑 → 0 < 𝐴)
 
Theoremnconstwlpolem 13677* Lemma for nconstwlpo 13678. (Contributed by Jim Kingdon, 23-Jul-2024.)
(𝜑𝐹:ℝ⟶ℤ)    &   (𝜑 → (𝐹‘0) = 0)    &   ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ≠ 0)    &   (𝜑𝐺:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺𝑖))       (𝜑 → (∀𝑦 ∈ ℕ (𝐺𝑦) = 0 ∨ ¬ ∀𝑦 ∈ ℕ (𝐺𝑦) = 0))
 
Theoremnconstwlpo 13678* Existence of a certain non-constant function from reals to integers implies ω ∈ WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.)
(𝜑𝐹:ℝ⟶ℤ)    &   (𝜑 → (𝐹‘0) = 0)    &   ((𝜑𝑥 ∈ ℝ+) → (𝐹𝑥) ≠ 0)       (𝜑 → ω ∈ WOmni)
 
Theoremneapmkvlem 13679* Lemma for neapmkv 13680. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))    &   ((𝜑𝐴 ≠ 1) → 𝐴 # 1)       (𝜑 → (¬ ∀𝑥 ∈ ℕ (𝐹𝑥) = 1 → ∃𝑥 ∈ ℕ (𝐹𝑥) = 0))
 
Theoremneapmkv 13680* If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.)
(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥𝑦𝑥 # 𝑦) → ω ∈ Markov)
 
11.3.7  Supremum and infimum
 
Theoremsupfz 13681 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
(𝑁 ∈ (ℤ𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)
 
Theoreminffz 13682 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
(𝑁 ∈ (ℤ𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)
 
11.3.8  Circle constant
 
Theoremtaupi 13683 Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
τ = (2 · π)
 
11.4  Mathbox for Mykola Mostovenko
 
Theoremax1hfs 13684 Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)
(𝜑 → (𝜑𝜑))
 
11.5  Mathbox for David A. Wheeler
 
11.5.1  Testable propositions
 
Theoremdftest 13685 A proposition is testable iff its negative or double-negative is true. See Chapter 2 [Moschovakis] p. 2.

We do not formally define testability with a new token, but instead use DECID ¬ before the formula in question. For example, DECID ¬ 𝑥 = 𝑦 corresponds to "𝑥 = 𝑦 is testable". (Contributed by David A. Wheeler, 13-Aug-2018.) For statements about testable propositions, search for the keyword "testable" in the comments of statements, for instance using the Metamath command "MM> SEARCH * "testable" / COMMENTS". (New usage is discouraged.)

(DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
 
11.5.2  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 𝑥𝜑𝜓 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(𝜑𝜓), and when restricted (applied to a class) we allow ∀!𝑥𝐴𝜑. 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.

 
Syntaxwalsi 13686 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(𝜑𝜓)
 
Syntaxwalsc 13687 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 ∀!𝑥𝐴𝜑
 
Definitiondf-alsi 13688 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(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑))
 
Definitiondf-alsc 13689 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.)
(∀!𝑥𝐴𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∃𝑥 𝑥𝐴))
 
Theoremalsconv 13690 There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
(∀!𝑥(𝑥𝐴𝜑) ↔ ∀!𝑥𝐴𝜑)
 
Theoremalsi1d 13691 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(𝜓𝜒))       (𝜑 → ∀𝑥(𝜓𝜒))
 
Theoremalsi2d 13692 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(𝜓𝜒))       (𝜑 → ∃𝑥𝜓)
 
Theoremalsc1d 13693 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 → ∀!𝑥𝐴𝜓)       (𝜑 → ∀𝑥𝐴 𝜓)
 
Theoremalsc2d 13694 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 → ∀!𝑥𝐴𝜓)       (𝜑 → ∃𝑥 𝑥𝐴)
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