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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lpowlpo 7201 | LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7200. There is an analogue in terms of analytic omniscience principles at tridceq 15292. (Contributed by Jim Kingdon, 24-Jul-2024.) |
⊢ (ω ∈ Omni → ω ∈ WOmni) | ||
Theorem | enwomnilem 7202 | Lemma for enwomni 7203. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni)) | ||
Theorem | enwomni 7203 | Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either ω ∈ WOmni or ℕ0 ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6459 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni ↔ 𝐵 ∈ WOmni)) | ||
Theorem | nninfdcinf 7204* | The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.) |
⊢ (𝜑 → ω ∈ WOmni) & ⊢ (𝜑 → 𝑁 ∈ ℕ∞) ⇒ ⊢ (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o)) | ||
Theorem | nninfwlporlemd 7205* | Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.) |
⊢ (𝜑 → 𝑋:ω⟶2o) & ⊢ (𝜑 → 𝑌:ω⟶2o) & ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)) ⇒ ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o))) | ||
Theorem | nninfwlporlem 7206* | Lemma for nninfwlpor 7207. The result. (Contributed by Jim Kingdon, 7-Dec-2024.) |
⊢ (𝜑 → 𝑋:ω⟶2o) & ⊢ (𝜑 → 𝑌:ω⟶2o) & ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅)) & ⊢ (𝜑 → ω ∈ WOmni) ⇒ ⊢ (𝜑 → DECID 𝑋 = 𝑌) | ||
Theorem | nninfwlpor 7207* | The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.) |
⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦) | ||
Theorem | nninfwlpoimlemg 7208* | Lemma for nninfwlpoim 7211. (Contributed by Jim Kingdon, 8-Dec-2024.) |
⊢ (𝜑 → 𝐹:ω⟶2o) & ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o)) ⇒ ⊢ (𝜑 → 𝐺 ∈ ℕ∞) | ||
Theorem | nninfwlpoimlemginf 7209* | Lemma for nninfwlpoim 7211. (Contributed by Jim Kingdon, 8-Dec-2024.) |
⊢ (𝜑 → 𝐹:ω⟶2o) & ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o)) ⇒ ⊢ (𝜑 → (𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)) | ||
Theorem | nninfwlpoimlemdc 7210* | Lemma for nninfwlpoim 7211. (Contributed by Jim Kingdon, 8-Dec-2024.) |
⊢ (𝜑 → 𝐹:ω⟶2o) & ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o)) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦) ⇒ ⊢ (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o) | ||
Theorem | nninfwlpoim 7211* | Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.) |
⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ω ∈ WOmni) | ||
Theorem | nninfwlpo 7212* | Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.) |
⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ω ∈ WOmni) | ||
Syntax | ccrd 7213 | Extend class definition to include the cardinal size function. |
class card | ||
Definition | df-card 7214* | Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.) |
⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}) | ||
Theorem | cardcl 7215* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On) | ||
Theorem | isnumi 7216 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) | ||
Theorem | finnum 7217 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | ||
Theorem | onenon 7218 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | ||
Theorem | cardval3ex 7219* | The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | ||
Theorem | oncardval 7220* | The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) | ||
Theorem | cardonle 7221 | The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) | ||
Theorem | card0 7222 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
⊢ (card‘∅) = ∅ | ||
Theorem | carden2bex 7223* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵)) | ||
Theorem | pm54.43 7224 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o)) | ||
Theorem | pr2nelem 7225 | Lemma for pr2ne 7226. (Contributed by FL, 17-Aug-2008.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | pr2ne 7226 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | exmidonfinlem 7227* | Lemma for exmidonfin 7228. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ⇒ ⊢ (ω = (On ∩ Fin) → DECID 𝜑) | ||
Theorem | exmidonfin 7228 | If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6904 and nnon 4630. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
⊢ (ω = (On ∩ Fin) → EXMID) | ||
Theorem | en2eleq 7229 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) | ||
Theorem | en2other2 7230 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋) | ||
Theorem | dju1p1e2 7231 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊢ (1o ⊔ 1o) ≈ 2o | ||
Theorem | infpwfidom 7232 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) | ||
Theorem | exmidfodomrlemeldju 7233 | Lemma for exmidfodomr 7238. A variant of djur 7102. (Contributed by Jim Kingdon, 2-Jul-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 1o) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) ⇒ ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) | ||
Theorem | exmidfodomrlemreseldju 7234 | Lemma for exmidfodomrlemrALT 7237. A variant of eldju 7101. (Contributed by Jim Kingdon, 9-Jul-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 1o) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) ⇒ ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅))) | ||
Theorem | exmidfodomrlemim 7235* | Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)) | ||
Theorem | exmidfodomrlemr 7236* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊢ (∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦) → EXMID) | ||
Theorem | exmidfodomrlemrALT 7237* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7236. In particular, this proof uses eldju 7101 instead of djur 7102 and avoids djulclb 7088. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.) |
⊢ (∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦) → EXMID) | ||
Theorem | exmidfodomr 7238* | Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊢ (EXMID ↔ ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦)) | ||
Syntax | wac 7239 | Formula for an abbreviation of the axiom of choice. |
wff CHOICE | ||
Definition | df-ac 7240* |
The expression CHOICE will be used as a
readable shorthand for any
form of the axiom of choice; all concrete forms are long, cryptic, have
dummy variables, or all three, making it useful to have a short name.
Similar to the Axiom of Choice (first form) of [Enderton] p. 49.
There are some decisions about how to write this definition especially around whether ax-setind 4557 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.) |
⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) | ||
Theorem | acfun 7241* | A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.) |
⊢ (𝜑 → CHOICE) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) | ||
Theorem | exmidaclem 7242* | Lemma for exmidac 7243. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})} & ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})} & ⊢ 𝐶 = {𝐴, 𝐵} ⇒ ⊢ (CHOICE → EXMID) | ||
Theorem | exmidac 7243 | The axiom of choice implies excluded middle. See acexmid 5899 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.) |
⊢ (CHOICE → EXMID) | ||
Theorem | endjudisj 7244 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) | ||
Theorem | djuen 7245 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷)) | ||
Theorem | djuenun 7246 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ∪ 𝐷)) | ||
Theorem | dju1en 7247 | Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴) | ||
Theorem | dju0en 7248 | Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) | ||
Theorem | xp2dju 7249 | Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | ||
Theorem | djucomen 7250 | Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) | ||
Theorem | djuassen 7251 | Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶))) | ||
Theorem | xpdjuen 7252 | Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶))) | ||
Theorem | djudoml 7253 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵)) | ||
Theorem | djudomr 7254 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ≼ (𝐴 ⊔ 𝐵)) | ||
Theorem | exmidontriimlem1 7255 | Lemma for exmidontriim 7259. A variation of r19.30dc 2637. (Contributed by Jim Kingdon, 12-Aug-2024.) |
⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | exmidontriimlem2 7256* | Lemma for exmidontriim 7259. (Contributed by Jim Kingdon, 12-Aug-2024.) |
⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → EXMID) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑦 ∈ 𝐵 𝑦 ∈ 𝐴)) | ||
Theorem | exmidontriimlem3 7257* | Lemma for exmidontriim 7259. What we get to do based on induction on both 𝐴 and 𝐵. (Contributed by Jim Kingdon, 10-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → EXMID) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) & ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
Theorem | exmidontriimlem4 7258* | Lemma for exmidontriim 7259. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → EXMID) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
Theorem | exmidontriim 7259* | Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.) |
⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | pw1on 7260 | The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.) |
⊢ 𝒫 1o ∈ On | ||
Theorem | pw1dom2 7261 | The power set of 1o dominates 2o. Also see pwpw0ss 3822 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.) |
⊢ 2o ≼ 𝒫 1o | ||
Theorem | pw1ne0 7262 | The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1o ≠ ∅ | ||
Theorem | pw1ne1 7263 | The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1o ≠ 1o | ||
Theorem | pw1ne3 7264 | The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1o ≠ 3o | ||
Theorem | pw1nel3 7265 | Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o) | ||
Theorem | sucpw1ne3 7266 | Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ (¬ EXMID → suc 𝒫 1o ≠ 3o) | ||
Theorem | sucpw1nel3 7267 | The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ ¬ suc 𝒫 1o ∈ 3o | ||
Theorem | 3nelsucpw1 7268 | Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ ¬ 3o ∈ suc 𝒫 1o | ||
Theorem | sucpw1nss3 7269 | Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.) |
⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o) | ||
Theorem | 3nsssucpw1 7270 | Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.) |
⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o) | ||
Theorem | onntri35 7271* |
Double negated ordinal trichotomy.
There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), (3) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (4) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7272), (3) implies (5) (onntri35 7271), (5) implies (1) (onntri51 7274), (2) implies (4) (onntri24 7276), (4) implies (5) (onntri45 7275), and (5) implies (2) (onntri52 7278). Another way of stating this is that EXMID is equivalent to trichotomy, either the 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 or the 𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥 form, as shown in exmidontri 7273 and exmidontri2or 7277, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7279 and (4) by onntri2or 7280. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID) | ||
Theorem | onntri13 7272 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | exmidontri 7273* | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | onntri51 7274* | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | onntri45 7275* | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID) | ||
Theorem | onntri24 7276 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
Theorem | exmidontri2or 7277* | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
Theorem | onntri52 7278* | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
Theorem | onntri3or 7279* | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | onntri2or 7280* | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
Syntax | wap 7281 | Apartness predicate symbol. |
wff 𝑅 Ap 𝐴 | ||
Definition | df-pap 7282* | Apartness predicate. A relation 𝑅 is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.) |
⊢ (𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧))))) | ||
Syntax | wtap 7283 | Tight apartness predicate symbol. |
wff 𝑅 TAp 𝐴 | ||
Definition | df-tap 7284* | Tight apartness predicate. A relation 𝑅 is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.) |
⊢ (𝑅 TAp 𝐴 ↔ (𝑅 Ap 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))) | ||
Theorem | dftap2 7285* | Tight apartness with the apartness properties from df-pap 7282 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.) |
⊢ (𝑅 TAp 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)))) | ||
Theorem | tapeq1 7286 | Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.) |
⊢ (𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴)) | ||
Theorem | tapeq2 7287 | Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.) |
⊢ (𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵)) | ||
Theorem | netap 7288* | Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴) | ||
Theorem | 2onetap 7289* | Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.) |
⊢ {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o | ||
Theorem | 2oneel 7290* | ∅ and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.) |
⊢ 〈∅, 1o〉 ∈ {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} | ||
Theorem | 2omotaplemap 7291* | Lemma for 2omotap 7293. (Contributed by Jim Kingdon, 6-Feb-2025.) |
⊢ (¬ ¬ 𝜑 → {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o) | ||
Theorem | 2omotaplemst 7292* | Lemma for 2omotap 7293. (Contributed by Jim Kingdon, 6-Feb-2025.) |
⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑) | ||
Theorem | 2omotap 7293 | If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.) |
⊢ (∃*𝑟 𝑟 TAp 2o → EXMID) | ||
Theorem | exmidapne 7294* | Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.) |
⊢ (EXMID → (𝑅 TAp 𝐴 ↔ 𝑅 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})) | ||
Theorem | exmidmotap 7295* | The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.) |
⊢ (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥) | ||
We have already introduced the full Axiom of Choice df-ac 7240 but since it implies excluded middle as shown at exmidac 7243, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle. | ||
Syntax | wacc 7296 | Formula for an abbreviation of countable choice. |
wff CCHOICE | ||
Definition | df-cc 7297* | The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7240 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.) |
⊢ (CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))) | ||
Theorem | ccfunen 7298* | Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → 𝐴 ≈ ω) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) | ||
Theorem | cc1 7299* | Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
⊢ (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧)) | ||
Theorem | cc2lem 7300* | Lemma for cc2 7301. (Contributed by Jim Kingdon, 27-Apr-2024.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → 𝐹 Fn ω) & ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥)) & ⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛))) & ⊢ 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴‘𝑛)))) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛))) |
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