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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfofr 6801* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑥𝑅       𝑥𝑟 𝑅

Theoremoffval 6802* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ (𝐶𝑅𝐷)))

Theoremofrfval 6803* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)    &   ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)       (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))

Theoremofval 6804 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)       ((𝜑𝑋𝑆) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Theoremofrval 6805 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)    &   ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)       ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)

Theoremoffn 6806 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝑆       (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝑆)

Theoremoffval2f 6807* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
𝑥𝜑    &   𝑥𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))

Theoremofmresval 6808 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
(𝜑𝐹𝐴)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹𝑓 𝑅𝐺))

Theoremfnfvof 6809 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))

Theoremoff 6810* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶       (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)

Theoremofres 6811 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝐴𝐵) = 𝐶       (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))

Theoremoffval2 6812* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))

Theoremofrfval2 6813* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐶𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))    &   (𝜑𝐺 = (𝑥𝐴𝐶))       (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝐴 𝐵𝑅𝐶))

Theoremofmpteq 6814* Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
((𝐴𝑉 ∧ (𝑥𝐴𝐵) Fn 𝐴 ∧ (𝑥𝐴𝐶) Fn 𝐴) → ((𝑥𝐴𝐵) ∘𝑓 𝑅(𝑥𝐴𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))

Theoremofco 6815 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐻:𝐷𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷𝑋)    &   (𝐴𝐵) = 𝐶       (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))

Theoremoffveq 6816* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐻 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)    &   ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)    &   ((𝜑𝑥𝐴) → (𝐵𝑅𝐶) = (𝐻𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = 𝐻)

Theoremoffveqb 6817* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   (𝜑𝐻 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)    &   ((𝜑𝑥𝐴) → (𝐺𝑥) = 𝐶)       (𝜑 → (𝐻 = (𝐹𝑓 𝑅𝐺) ↔ ∀𝑥𝐴 (𝐻𝑥) = (𝐵𝑅𝐶)))

Theoremofc1 6818 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)       ((𝜑𝑋𝐴) → (((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))

Theoremofc2 6819 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)       ((𝜑𝑋𝐴) → ((𝐹𝑓 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))

Theoremofc12 6820 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))

Theoremcaofref 6821* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑𝑥𝑆) → 𝑥𝑅𝑥)       (𝜑𝐹𝑟 𝑅𝐹)

Theoremcaofinvl 6822* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝑁:𝑆𝑆)    &   (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))    &   ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)       (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))

Theoremcaofid0l 6823* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)

Theoremcaofid0r 6824* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝑥)       (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = 𝐹)

Theoremcaofid1 6825* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)       (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))

Theoremcaofid2 6826* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))

Theoremcaofcom 6827* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))

Theoremcaofrss 6828* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))       (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))

Theoremcaofass 6829* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))       (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)))

Theoremcaoftrn 6830* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))       (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))

Theoremcaofdi 6831* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐾)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))       (𝜑 → (𝐹𝑓 𝑇(𝐺𝑓 𝑅𝐻)) = ((𝐹𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹𝑓 𝑇𝐻)))

Theoremcaofdir 6832* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐾)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))       (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)))

Theoremcaonncan 6833* Transfer nncan 10189-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑𝐼𝑉)    &   (𝜑𝐴:𝐼𝑆)    &   (𝜑𝐵:𝐼𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)       (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = 𝐵)

2.3.20  Proper subset relation

Syntaxcrpss 6834 Extend class notation to include the reified proper subset relation.
class []

Definitiondf-rpss 6835* Define a relation which corresponds to proper subsethood df-pss 3556 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6840. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}

Theoremrelrpss 6836 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Rel []

Theorembrrpssg 6837 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))

Theorembrrpss 6838 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐵 ∈ V       (𝐴 [] 𝐵𝐴𝐵)

Theoremporpss 6839 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] Po 𝐴

Theoremsorpss 6840* 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 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))

Theoremsorpssi 6841 Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))

Theoremsorpssun 6842 A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)

Theoremsorpssin 6843 A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)

Theoremsorpssuni 6844* In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))

Theoremsorpssint 6845* In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))

Theoremsorpsscmpl 6846* 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 ∣ (𝐴𝑢) ∈ 𝑌})

2.4  ZF Set Theory - add the Axiom of Union

2.4.1  Introduce the Axiom of Union

Axiomax-un 6847* 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 6849 states that the union itself exists. A version with the standard abbreviation for union is uniex2 6850. A version using class notation is uniex 6851.

The union of a class df-uni 4373 should not be confused with the union of two classes df-un 3545. Their relationship is shown in unipr 4385. (Contributed by NM, 23-Dec-1993.)

𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)

Theoremzfun 6848* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Theoremaxun2 6849* A variant of the Axiom of Union ax-un 6847. 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.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))

Theoremuniex2 6850* 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

Theoremuniex 6851 The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3180), 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

Theoremvuniex 6852 The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
𝑥 ∈ V

Theoremuniexg 6853 The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴𝑉 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.)
(𝐴𝑉 𝐴 ∈ V)

Theoremunex 6854 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       (𝐴𝐵) ∈ V

Theoremtpex 6855 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
{𝐴, 𝐵, 𝐶} ∈ V

Theoremunexb 6856 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Theoremunexg 6857 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoremxpexg 6858 The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 7052. (Contributed by NM, 14-Aug-1994.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)

Theorem3xpexg 6859 The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
(𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)

Theoremxpex 6860 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

Theoremsqxpexg 6861 The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
(𝐴𝑉 → (𝐴 × 𝐴) ∈ V)

Theoremsnnex 6862* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)
{𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Theoremdifex2 6863 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.)
(𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))

Theoremdifsnexi 6864 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)

Theoremuniuni 6865* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥 = 𝑦𝑦𝐴)}

Theoremuniexb 6866 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)

Theorempwexb 6867 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)

Theoremeldifpw 6868 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
𝐶 ∈ V       ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))

Theoremelpwun 6869 Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
𝐶 ∈ V       (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)

Theoremiunpw 6870* 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       (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)

Theoremfr3nr 6871 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 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))

Theoremepne3 6872 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 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))

Theoremdfwe2 6873* 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 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))

2.4.2  Ordinals (continued)

Theoremordon 6874 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

Theoremepweon 6875 The epsilon 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

Theoremonprc 6876 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 6874), 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

Theoremssorduni 6877 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)

Theoremssonuni 6878 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
(𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))

Theoremssonunii 6879 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)

Theoremordeleqon 6880 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))

Theoremordsson 6881 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 𝐴𝐴 ⊆ On)

Theoremonss 6882 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)

Theorempredon 6883 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)

Theoremssonprc 6884 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))

Theoremonuni 6885 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)

Theoremorduni 6886 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)

Theoremonint 6887 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 ≠ ∅) → 𝐴𝐴)

Theoremonint0 6888 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))

Theoremonssmin 6889* 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 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremonminesb 6890 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)

Theoremonminsb 6891 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.)
𝑥𝜓    &   (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))       (∃𝑥 ∈ On 𝜑𝜓)

Theoremoninton 6892 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)

Theoremonintrab 6893 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)

Theoremonintrab2 6894 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)

Theoremonnmin 6895 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)

Theoremonnminsb 6896* 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 → (𝜑𝜓))       (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))

Theoremoneqmin 6897* 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 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))

Theorembm2.5ii 6898* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})

Theoremonminex 6899* If a wff is true for an ordinal number, 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 → (𝜑𝜓))       (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))

Theoremsucon 6900 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On

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