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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fodjum 7101* | Lemma for fodjuomni 7104 and fodjumkv 7115. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) & ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) & ⊢ (𝜑 → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) ⇒ ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | ||
Theorem | fodju0 7102* | Lemma for fodjuomni 7104 and fodjumkv 7115. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) & ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) & ⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) ⇒ ⊢ (𝜑 → 𝐴 = ∅) | ||
Theorem | fodjuomnilemres 7103* | Lemma for fodjuomni 7104. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.) |
⊢ (𝜑 → 𝑂 ∈ Omni) & ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) & ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) ⇒ ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅)) | ||
Theorem | fodjuomni 7104* | A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.) |
⊢ (𝜑 → 𝑂 ∈ Omni) & ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) ⇒ ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅)) | ||
Theorem | ctssexmid 7105* | The decidability condition in ctssdc 7069 is needed. More specifically, ctssdc 7069 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.) |
⊢ ((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o)) & ⊢ ω ∈ Omni ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Syntax | cmarkov 7106 | Extend class definition to include the class of Markov sets. |
class Markov | ||
Definition | df-markov 7107* |
A Markov set is one where if a predicate (here represented by a function
𝑓) on that set does not hold (where
hold means is equal to 1o)
for all elements, then there exists an element where it fails (is equal
to ∅). Generalization of definition 2.5
of [Pierik], p. 9.
In particular, ω ∈ Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.) |
⊢ Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅))} | ||
Theorem | ismkv 7108* | The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))) | ||
Theorem | ismkvmap 7109* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))) | ||
Theorem | ismkvnex 7110* | The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) | ||
Theorem | omnimkv 7111 | An omniscient set is Markov. In particular, the case where 𝐴 is ω means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.) |
⊢ (𝐴 ∈ Omni → 𝐴 ∈ Markov) | ||
Theorem | exmidmp 7112 | Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.) |
⊢ (EXMID → ω ∈ Markov) | ||
Theorem | mkvprop 7113* | Markov's Principle expressed in terms of propositions (or more precisely, the 𝐴 = ω case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.) |
⊢ ((𝐴 ∈ Markov ∧ ∀𝑛 ∈ 𝐴 DECID 𝜑 ∧ ¬ ∀𝑛 ∈ 𝐴 ¬ 𝜑) → ∃𝑛 ∈ 𝐴 𝜑) | ||
Theorem | fodjumkvlemres 7114* | Lemma for fodjumkv 7115. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.) |
⊢ (𝜑 → 𝑀 ∈ Markov) & ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵)) & ⊢ 𝑃 = (𝑦 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) ⇒ ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)) | ||
Theorem | fodjumkv 7115* | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
⊢ (𝜑 → 𝑀 ∈ Markov) & ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)) | ||
Theorem | enmkvlem 7116 | Lemma for enmkv 7117. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov)) | ||
Theorem | enmkv 7117 | Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either ω ∈ Markov or ℕ0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6389 says that 2o = {∅, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov ↔ 𝐵 ∈ Markov)) | ||
Syntax | cwomni 7118 | Extend class definition to include the class of weakly omniscient sets. |
class WOmni | ||
Definition | df-womni 7119* |
A weakly omniscient set is one where we can decide whether a predicate
(here represented by a function 𝑓) holds (is equal to 1o) for
all elements or not. Generalization of definition 2.4 of [Pierik],
p. 9.
In particular, ω ∈ WOmni is known as the Weak Limited Principle of Omniscience (WLPO). The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.) |
⊢ WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)} | ||
Theorem | iswomni 7120* | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) | ||
Theorem | iswomnimap 7121* | The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) | ||
Theorem | omniwomnimkv 7122 | A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = ω says that LPO ↔ WLPO ∧ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.) |
⊢ (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov)) | ||
Theorem | lpowlpo 7123 | LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7122. There is an analogue in terms of analytic omniscience principles at tridceq 13776. (Contributed by Jim Kingdon, 24-Jul-2024.) |
⊢ (ω ∈ Omni → ω ∈ WOmni) | ||
Theorem | enwomnilem 7124 | Lemma for enwomni 7125. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni)) | ||
Theorem | enwomni 7125 | 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 6389 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)) | ||
Syntax | ccrd 7126 | Extend class definition to include the cardinal size function. |
class card | ||
Definition | df-card 7127* | 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 7128* | 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 7129 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) | ||
Theorem | finnum 7130 | 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 7131 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | ||
Theorem | cardval3ex 7132* | The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | ||
Theorem | oncardval 7133* | 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 7134 | 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 7135 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
⊢ (card‘∅) = ∅ | ||
Theorem | carden2bex 7136* | 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 7137 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o)) | ||
Theorem | pr2nelem 7138 | Lemma for pr2ne 7139. (Contributed by FL, 17-Aug-2008.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | pr2ne 7139 | 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 7140* | Lemma for exmidonfin 7141. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ⇒ ⊢ (ω = (On ∩ Fin) → DECID 𝜑) | ||
Theorem | exmidonfin 7141 | 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 6829 and nnon 4581. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
⊢ (ω = (On ∩ Fin) → EXMID) | ||
Theorem | en2eleq 7142 | 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 7143 | 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 7144 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
⊢ (1o ⊔ 1o) ≈ 2o | ||
Theorem | infpwfidom 7145 | 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 7146 | Lemma for exmidfodomr 7151. A variant of djur 7025. (Contributed by Jim Kingdon, 2-Jul-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 1o) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) ⇒ ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) | ||
Theorem | exmidfodomrlemreseldju 7147 | Lemma for exmidfodomrlemrALT 7150. A variant of eldju 7024. (Contributed by Jim Kingdon, 9-Jul-2022.) |
⊢ (𝜑 → 𝐴 ⊆ 1o) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) ⇒ ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅))) | ||
Theorem | exmidfodomrlemim 7148* | 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 7149* | 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 7150* | 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 7149. In particular, this proof uses eldju 7024 instead of djur 7025 and avoids djulclb 7011. (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 7151* | 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 7152 | Formula for an abbreviation of the axiom of choice. |
wff CHOICE | ||
Definition | df-ac 7153* |
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 4508 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 7154* | 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 7155* | Lemma for exmidac 7156. 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 7156 | The axiom of choice implies excluded middle. See acexmid 5835 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 7157 | 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 7158 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷)) | ||
Theorem | djuenun 7159 | 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 7160 | 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 7161 | 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 7162 | 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 7163 | 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 7164 | 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 7165 | 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 7166 | 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 7167 | 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 7168 | Lemma for exmidontriim 7172. A variation of r19.30dc 2611. (Contributed by Jim Kingdon, 12-Aug-2024.) |
⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | exmidontriimlem2 7169* | Lemma for exmidontriim 7172. (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 7170* | Lemma for exmidontriim 7172. 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 7171* | Lemma for exmidontriim 7172. 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 7172* | 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 7173 | The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.) |
⊢ 𝒫 1o ∈ On | ||
Theorem | pw1dom2 7174 | The power set of 1o dominates 2o. Also see pwpw0ss 3778 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.) |
⊢ 2o ≼ 𝒫 1o | ||
Theorem | pw1ne0 7175 | The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1o ≠ ∅ | ||
Theorem | pw1ne1 7176 | The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1o ≠ 1o | ||
Theorem | pw1ne3 7177 | The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.) |
⊢ 𝒫 1o ≠ 3o | ||
Theorem | pw1nel3 7178 | 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 7179 | 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 7180 | 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 7181 | 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 7182 | 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 7183 | 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 7184* |
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 7185), (3) implies (5) (onntri35 7184), (5) implies (1) (onntri51 7187), (2) implies (4) (onntri24 7189), (4) implies (5) (onntri45 7188), and (5) implies (2) (onntri52 7191). 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 7186 and exmidontri2or 7190, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7192 and (4) by onntri2or 7193. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID) | ||
Theorem | onntri13 7185 | 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 7186* | 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 7187* | 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 7188* | 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 7189 | 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 7190* | 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 7191* | 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 7192* | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | onntri2or 7193* | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
We have already introduced the full Axiom of Choice df-ac 7153 but since it implies excluded middle as shown at exmidac 7156, 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 7194 | Formula for an abbreviation of countable choice. |
wff CCHOICE | ||
Definition | df-cc 7195* | The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7153 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.) |
⊢ (CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))) | ||
Theorem | ccfunen 7196* | 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 7197* | 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 7198* | Lemma for cc2 7199. (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))) | ||
Theorem | cc2 7199* | Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → 𝐹 Fn ω) & ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛))) | ||
Theorem | cc3 7200* | Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.) |
⊢ (𝜑 → CCHOICE) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V) & ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹) & ⊢ (𝜑 → 𝑁 ≈ ω) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)) |
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