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Theorem List for Intuitionistic Logic Explorer - 13201-13283   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-inf2vnlem4 13201* Lemma for bj-inf2vn2 13203. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))

Theorembj-inf2vn 13202* A sufficient condition for ω to be a set. See bj-inf2vn2 13203 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       (𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Theorembj-inf2vn2 13203* A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13202. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Axiomax-inf2 13204* Another axiom of infinity in a constructive setting (see ax-infvn 13169). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))

Theorembj-omex2 13205 Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13169 (see bj-2inf 13166 for the equivalence of the latter with bj-omex 13170). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ∈ V

Theorembj-nn0sucALT 13206* Alternate proof of bj-nn0suc 13192, also constructive but from ax-inf2 13204, hence requiring ax-bdsetind 13196. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

11.2.10.2  Full induction

In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.

Theorembj-findis 13207* 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 13175 for a bounded version not requiring ax-setind 4452. See finds 4514 for a proof in IZF. From this version, it is easy to prove of finds 4514, finds2 4515, finds1 4516. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝑥𝜒    &   𝑥𝜃    &   (𝑥 = ∅ → (𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜃𝜑))       ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)

Theorembj-findisg 13208* Version of bj-findis 13207 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13207 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝑥𝜒    &   𝑥𝜃    &   (𝑥 = ∅ → (𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜃𝜑))    &   𝑥𝐴    &   𝑥𝜏    &   (𝑥 = 𝐴 → (𝜑𝜏))       ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (𝐴 ∈ ω → 𝜏))

Theorembj-findes 13209 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13207 for explanations. From this version, it is easy to prove findes 4517. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
(([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

11.2.11  CZF: Strong collection

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

Axiomax-strcoll 13210* 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(∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

Theoremstrcoll2 13211* Version of ax-strcoll 13210 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
(∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

Theoremstrcollnft 13212* Closed form of strcollnf 13213. Version of ax-strcoll 13210 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
(∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))

Theoremstrcollnf 13213* Version of ax-strcoll 13210 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

TheoremstrcollnfALT 13214* Alternate proof of strcollnf 13213, not using strcollnft 13212. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

11.2.12  CZF: Subset collection

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

Axiomax-sscoll 13215* 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.)
𝑎𝑏𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))

Theoremsscoll2 13216* Version of ax-sscoll 13215 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))

11.2.13  Real numbers

Axiomax-ddkcomp 13217 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 the axiom ax-ddkcomp 13217 should be used in place of construction specific results. In particular, axcaucvg 7715 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

Theoremel2oss1o 13218 Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13219. (Contributed by Jim Kingdon, 8-Aug-2022.)
(𝐴 ∈ 2o𝐴 ⊆ 1o)

Theoremss1oel2o 13219 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4121 which more directly illustrates the contrast with el2oss1o 13218. (Contributed by Jim Kingdon, 8-Aug-2022.)
(EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))

Theorempw1dom2 13220 The power set of 1o dominates 2o. Also see pwpw0ss 3731 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
2o ≼ 𝒫 1o

Theoremnnti 13221 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)))

11.3.2  The power set of a singleton

Theorempwtrufal 13222 A subset of the singleton {∅} cannot be anything other than or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4121. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4120), 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 13223* 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 13224* 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 13225* 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 13226* 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)

11.3.3  Omniscience of NN+oo

Theorem0nninf 13227 The zero element of (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.)
(ω × {∅}) ∈ ℕ

Theoremnninff 13228 An element of is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝐴 ∈ ℕ𝐴:ω⟶2o)

Theoremnnsf 13229* 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 13230* 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 13231* 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 ↦ ∅))

Theoremnninfalllemn 13232* Lemma for nninfall 13234. Mapping of a natural number to an element of . (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑁 ∈ ω)    &   (𝜑 → ∀𝑥𝑁 (𝑃𝑥) = 1o)    &   (𝜑 → (𝑃𝑁) = ∅)       (𝜑𝑃 = (𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅)))

Theoremnninfalllem1 13233* Lemma for nninfall 13234. (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 13234* 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)

Theoremnninfex 13235 is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
∈ V

Theoremnninfsellemdc 13236* Lemma for nninfself 13239. 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 13237* Lemma for nninfself 13239. (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 13238* Lemma for nninfself 13239. (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 13239* 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 13240* Lemma for nninfsel 13243. (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 13241* Lemma for nninfsel 13243. (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 13242* Lemma for nninfsel 13243. (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 13243* 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 13244* Lemma for nninfomni 13245. (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 13245 is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
∈ Omni

Theoremnninffeq 13246* Equality of two functions on which agree at every integer and at the point at infinity. From an online post by Martin Escardo. (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 13247* Lemma for exmidsbthr 13248. (Contributed by Jim Kingdon, 11-Aug-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)

Theoremexmidsbthr 13248* 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 13249* The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6855) 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-intuitionist proof at https://us.metamath.org/mpeuni/sbth.html 6855.

The reverse direction (exmidsbthr 13248) 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 13250 Lemma for sbthom 13251. (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 13251 Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 13249 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 13252* 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 10981 (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 13253* 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 13254 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 13255* Lemma for isomninn 13256. 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 13256* Omniscience stated in terms of natural numbers. Similar to isomnimap 7009 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 13257* Lemma for cvgcmp2n 13258. 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 13258* 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 ⇝ )

Theoremtrilpolemclim 13259* Lemma for trilpo 13266. 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 13260* Lemma for trilpo 13266. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)
(𝜑𝐹:ℕ⟶{0, 1})    &   𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹𝑖))       (𝜑𝐴 ∈ ℝ)

Theoremtrilpolemisumle 13261* Lemma for trilpo 13266. 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 13262* Lemma for trilpo 13266. 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 13263* Lemma for trilpo 13266. 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 13264* Lemma for trilpo 13266. 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 7012 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 13265* Lemma for trilpo 13266. 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 13266* 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 13264 (which means the sequence contains a zero), trilpolemeq1 13263 (which means the sequence is all ones), and trilpolemgt1 13262 (which is not possible). (Contributed by Jim Kingdon, 23-Aug-2023.)

(∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) → ω ∈ Omni)

Theoremapdifflemf 13267 Lemma for apdiff 13269. 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 13268 Lemma for apdiff 13269. (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 13269* 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)))))

11.3.6  Supremum and infimum

Theoremsupfz 13270 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 13271 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)

11.3.7  Circle constant

Theoremtaupi 13272 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 13273 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 13274 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 13275 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 13276 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 13277 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 13278 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 13279 There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
(∀!𝑥(𝑥𝐴𝜑) ↔ ∀!𝑥𝐴𝜑)

Theoremalsi1d 13280 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 13281 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 13282 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 13283 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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