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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | caofinvl 7201* | Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑁:𝑆⟶𝑆) & ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝐴 × {𝐵})) | ||
Theorem | caofid0l 7202* | Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝑥) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹) | ||
Theorem | caofid0r 7203* | Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝑥) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅(𝐴 × {𝐵})) = 𝐹) | ||
Theorem | caofid1 7204* | Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶})) | ||
Theorem | caofid2 7205* | Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐵𝑅𝑥) = 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶})) | ||
Theorem | caofcom 7206* | Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹)) | ||
Theorem | caofrss 7207* | Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 → 𝐹 ∘𝑟 𝑇𝐺)) | ||
Theorem | caofass 7208* | Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧))) ⇒ ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹 ∘𝑓 𝑂(𝐺 ∘𝑓 𝑃𝐻))) | ||
Theorem | caoftrn 7209* | Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧)) ⇒ ⊢ (𝜑 → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) → 𝐹 ∘𝑟 𝑈𝐻)) | ||
Theorem | caofdi 7210* | Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧))) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇(𝐺 ∘𝑓 𝑅𝐻)) = ((𝐹 ∘𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹 ∘𝑓 𝑇𝐻))) | ||
Theorem | caofdir 7211* | Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) & ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) & ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧))) ⇒ ⊢ (𝜑 → ((𝐺 ∘𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺 ∘𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻 ∘𝑓 𝑇𝐹))) | ||
Theorem | caonncan 7212* | Transfer nncan 10652-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐴:𝐼⟶𝑆) & ⊢ (𝜑 → 𝐵:𝐼⟶𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) ⇒ ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀(𝐴 ∘𝑓 𝑀𝐵)) = 𝐵) | ||
Syntax | crpss 7213 | Extend class notation to include the reified proper subset relation. |
class [⊊] | ||
Definition | df-rpss 7214* | Define a relation which corresponds to proper subsethood df-pss 3808 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 7219. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} | ||
Theorem | relrpss 7215 | The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ Rel [⊊] | ||
Theorem | brrpssg 7216 | The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵)) | ||
Theorem | brrpss 7217 | The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 [⊊] 𝐵 ↔ 𝐴 ⊊ 𝐵) | ||
Theorem | porpss 7218 | Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ [⊊] Po 𝐴 | ||
Theorem | sorpss 7219* | Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) | ||
Theorem | sorpssi 7220 | Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | ||
Theorem | sorpssun 7221 | A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.) |
⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴) | ||
Theorem | sorpssin 7222 | A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.) |
⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴) | ||
Theorem | sorpssuni 7223* | In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ∈ 𝑌)) | ||
Theorem | sorpssint 7224* | In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∩ 𝑌 ∈ 𝑌)) | ||
Theorem | sorpsscmpl 7225* | The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ ( [⊊] Or 𝑌 → [⊊] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}) | ||
Axiom | ax-un 7226* |
Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states
that a set 𝑦 exists that includes the union of a
given set 𝑥
i.e. the collection of all members of the members of 𝑥. The
variant axun2 7228 states that the union itself exists. A
version with the
standard abbreviation for union is uniex2 7229. A version using class
notation is uniex 7230.
The union of a class df-uni 4672 should not be confused with the union of two classes df-un 3797. Their relationship is shown in unipr 4684. (Contributed by NM, 23-Dec-1993.) |
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfun 7227* | Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) |
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axun2 7228* | A variant of the Axiom of Union ax-un 7226. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | ||
Theorem | uniex2 7229* | The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) |
⊢ ∃𝑦 𝑦 = ∪ 𝑥 | ||
Theorem | uniex 7230 | The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3409), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∪ 𝐴 ∈ V | ||
Theorem | vuniex 7231 | The union of a setvar is a set. (Contributed by BJ, 3-May-2021.) |
⊢ ∪ 𝑥 ∈ V | ||
Theorem | uniexg 7232 | The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | ||
Theorem | unex 7233 | The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ V | ||
Theorem | tpex 7234 | An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.) |
⊢ {𝐴, 𝐵, 𝐶} ∈ V | ||
Theorem | unexb 7235 | Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | unexg 7236 | A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | xpexg 7237 | The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 7438. (Contributed by NM, 14-Aug-1994.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) | ||
Theorem | xpexd 7238 | The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) | ||
Theorem | 3xpexg 7239 | The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.) |
⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V) | ||
Theorem | xpex 7240 | The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 × 𝐵) ∈ V | ||
Theorem | sqxpexg 7241 | The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | ||
Theorem | abnexg 7242* | Sufficient condition for a class abstraction to be a proper class. The class 𝐹 can be thought of as an expression in 𝑥 and the abstraction appearing in the statement as the class of values 𝐹 as 𝑥 varies through 𝐴. Assuming the antecedents, if that class is a set, then so is the "domain" 𝐴. The converse holds without antecedent, see abrexexg 7419. Note that the second antecedent ∀𝑥 ∈ 𝐴𝑥 ∈ 𝐹 cannot be translated to 𝐴 ⊆ 𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 7244 and pwnex 7245 respectively, proved from abnex 7243, which is a consequence of abnexg 7242 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.) |
⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V)) | ||
Theorem | abnex 7243* | Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7244 and pwnex 7245. See the comment of abnexg 7242. (Contributed by BJ, 2-May-2021.) |
⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V) | ||
Theorem | snnex 7244* | The class of all singletons is a proper class. See also pwnex 7245. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.) |
⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | ||
Theorem | pwnex 7245* | The class of all power sets is a proper class. See also snnex 7244. (Contributed by BJ, 2-May-2021.) |
⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V | ||
Theorem | difex2 7246 | If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V)) | ||
Theorem | difsnexi 7247 | If the difference of a class and a singleton is a set, the class itself is a set. (Contributed by AV, 15-Jan-2019.) |
⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) | ||
Theorem | uniuni 7248* | Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.) |
⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} | ||
Theorem | uniexr 7249 | Converse of the Axiom of Union. Note that it does not require ax-un 7226. (Contributed by NM, 11-Nov-2003.) |
⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V) | ||
Theorem | uniexb 7250 | The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | ||
Theorem | pwexr 7251 | Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5077. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V) | ||
Theorem | pwexb 7252 | The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | ||
Theorem | elpwpwel 7253 | A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.) |
⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | eldifpw 7254 | Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵)) | ||
Theorem | elpwun 7255 | Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) | ||
Theorem | iunpw 7256* | An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥) | ||
Theorem | fr3nr 7257 | A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
Theorem | epne3 7258 | A set well-founded by epsilon contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐵)) | ||
Theorem | dfwe2 7259* | Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
Theorem | epweon 7260 | The membership relation well-orders the class of ordinal numbers. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.) |
⊢ E We On | ||
Theorem | ordon 7261 | The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.) |
⊢ Ord On | ||
Theorem | onprc 7262 | No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 7261), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.) |
⊢ ¬ On ∈ V | ||
Theorem | ssorduni 7263 | The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | ||
Theorem | ssonuni 7264 | The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) | ||
Theorem | ssonunii 7265 | The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On) | ||
Theorem | ordeleqon 7266 | A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse. (Contributed by NM, 1-Jun-2003.) |
⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | ||
Theorem | ordsson 7267 | Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (Ord 𝐴 → 𝐴 ⊆ On) | ||
Theorem | onss 7268 | An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | ||
Theorem | predon 7269 | For an ordinal, the predecessor under E and On is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.) |
⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) | ||
Theorem | ssonprc 7270 | Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ (𝐴 ⊆ On → (𝐴 ∉ V ↔ ∪ 𝐴 = On)) | ||
Theorem | onuni 7271 | The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | ||
Theorem | orduni 7272 | The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.) |
⊢ (Ord 𝐴 → Ord ∪ 𝐴) | ||
Theorem | onint 7273 | The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | ||
Theorem | onint0 7274 | The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.) |
⊢ (𝐴 ⊆ On → (∩ 𝐴 = ∅ ↔ ∅ ∈ 𝐴)) | ||
Theorem | onssmin 7275* | A nonempty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) | ||
Theorem | onminesb 7276 | If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.) |
⊢ (∃𝑥 ∈ On 𝜑 → [∩ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑) | ||
Theorem | onminsb 7277 | If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝜑} → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ On 𝜑 → 𝜓) | ||
Theorem | oninton 7278 | The intersection of a nonempty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On) | ||
Theorem | onintrab 7279 | The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.) |
⊢ (∩ {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | ||
Theorem | onintrab2 7280 | An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.) |
⊢ (∃𝑥 ∈ On 𝜑 ↔ ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | ||
Theorem | onnmin 7281 | No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) |
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴) | ||
Theorem | onnminsb 7282* | An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. 𝜓 is the wff resulting from the substitution of 𝐴 for 𝑥 in wff 𝜑. (Contributed by NM, 9-Nov-2003.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓)) | ||
Theorem | oneqmin 7283* | A way to show that an ordinal number equals the minimum of a nonempty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.) |
⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))) | ||
Theorem | uniordint 7284* | The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
Theorem | onminex 7285* | If a wff is true for an ordinal number, then there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦 ∈ 𝑥 ¬ 𝜓)) | ||
Theorem | sucon 7286 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
⊢ suc On = On | ||
Theorem | sucexb 7287 | A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.) |
⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | ||
Theorem | sucexg 7288 | The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | ||
Theorem | sucex 7289 | The successor of a set is a set. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ suc 𝐴 ∈ V | ||
Theorem | onmindif2 7290 | The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. (Contributed by NM, 20-Nov-2003.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴})) | ||
Theorem | suceloni 7291 | The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | ||
Theorem | ordsuc 7292 | The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) |
⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | ||
Theorem | ordpwsuc 7293 | The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.) |
⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴) | ||
Theorem | onpwsuc 7294 | The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.) |
⊢ (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴) | ||
Theorem | sucelon 7295 | The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | ||
Theorem | ordsucss 7296 | The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | ||
Theorem | onpsssuc 7297 | An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) | ||
Theorem | ordelsuc 7298 | A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | ||
Theorem | onsucmin 7299* | The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥}) | ||
Theorem | ordsucelsuc 7300 | Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) |
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