P. 78 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7701-7800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ofreq 7701	Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆)
Theorem	ofexg 7702	A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → ( ∘f 𝑅 ↾ 𝐴) ∈ V)
Theorem	nfof 7703	Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥 ∘f 𝑅
Theorem	nfofr 7704	Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥 ∘r 𝑅
Theorem	ofrfvalg 7705*	Value of a relation applied to two functions. Originally part of ofrfval 7707, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5279. (Contributed by SN, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
Theorem	offval 7706*	Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷)))
Theorem	ofrfval 7707*	Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
Theorem	ofval 7708	Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Theorem	ofrval 7709	Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)
Theorem	offn 7710	The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝑆    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆)
Theorem	offun 7711	The function operation produces a function. (Contributed by SN, 23-Jul-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺))
Theorem	offval2f 7712*	The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
Theorem	ofmresval 7713	Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
⊢ (𝜑 → 𝐹 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘f 𝑅𝐺))
Theorem	fnfvof 7714	Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
Theorem	off 7715*	The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)
Theorem	ofres 7716	Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)))
Theorem	offval2 7717*	The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
Theorem	ofrfval2 7718*	The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶))
Theorem	offvalfv 7719*	The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
Theorem	ofmpteq 7720*	Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
Theorem	coof 7721*	The composition of a homomorphism with a function operation. (Contributed by SN, 20-May-2025.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)))
Theorem	ofco 7722	The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐻:𝐷⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))
Theorem	offveq 7723*	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ (𝜑 → 𝐻 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻)
Theorem	offveqb 7724*	Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ (𝜑 → 𝐻 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))
Theorem	ofc1 7725	Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶))
Theorem	ofc2 7726	Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))
Theorem	ofc12 7727	Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
Theorem	caofref 7728*	Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐹)
Theorem	caofinvl 7729*	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 7730*	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 7731*	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 7732*	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 7733*	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 7734*	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	caofidlcan 7735*	Transfer a cancellation/identity law to the function operation. (Contributed by SN, 16-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑥𝑅𝑦) = 𝑦 ↔ 𝑥 = 0 ))    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = 𝐺 ↔ 𝐹 = (𝐴 × { 0 })))
Theorem	caofrss 7736*	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 7737*	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 7738*	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 7739*	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 7740*	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 7741*	Transfer nncan 11538-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)
2.3.22  Proper subset relation
Syntax	crpss 7742	Extend class notation to include the reified proper subset relation.
class [⊊]
Definition	df-rpss 7743*	Define a relation which corresponds to proper subsethood df-pss 3971 on sets. This allows to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 7748. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
Theorem	relrpss 7744	The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ Rel [⊊]
Theorem	brrpssg 7745	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 7746	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 7747	Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ [⊊] Po 𝐴
Theorem	sorpss 7748*	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 7749	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 7750	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 7751	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 7752*	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 7753*	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 7754*	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})
2.4  ZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union
Axiom	ax-un 7755*	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 7757 states that the union itself exists. A version with the standard abbreviation for union is uniex2 7758. A version using class notation is uniex 7761. The union of a class df-uni 4908 should not be confused with the union of two classes df-un 3956. Their relationship is shown in unipr 4924. (Contributed by NM, 23-Dec-1993.)
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfun 7756*	Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) Use ax-un 7755 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axun2 7757*	A variant of the Axiom of Union ax-un 7755. 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 7758*	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	vuniex 7759	The union of a setvar is a set. (Contributed by BJ, 3-May-2021.) (Revised by BJ, 6-Apr-2024.)
⊢ ∪ 𝑥 ∈ V
Theorem	uniexg 7760	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	uniex 7761	The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3494), 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	uniexd 7762	Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ V)
Theorem	unexg 7763	The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) Prove unexg 7763 first and then unex 7764 and unexb 7767 from it. (Revised by BJ, 21-Jul-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	unex 7764	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	unexOLD 7765	Obsolete version of unex 7764 as of 21-Jul-2025. (Contributed by NM, 1-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ V
Theorem	tpex 7766	An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
⊢ {𝐴, 𝐵, 𝐶} ∈ V
Theorem	unexb 7767	Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	unexbOLD 7768	Obsolete version of unexb 7767 as of 21-Jul-2025. (Contributed by NM, 11-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	unexgOLD 7769	Obsolete version of unexg 7763 as of 21-Jul-2025. (Contributed by NM, 18-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	xpexg 7770	The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 8006. (Contributed by NM, 14-Aug-1994.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
Theorem	xpexd 7771	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 7772	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 7773	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	unexd 7774	The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
Theorem	sqxpexg 7775	The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
Theorem	abnexg 7776*	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 7985. 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 7778 and pwnex 7779 respectively, proved from abnex 7777, which is a consequence of abnexg 7776 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 7777*	Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7778 and pwnex 7779. See the comment of abnexg 7776. (Contributed by BJ, 2-May-2021.)
⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)
Theorem	snnex 7778*	The class of all singletons is a proper class. See also pwnex 7779. (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 7779*	The class of all power sets is a proper class. See also snnex 7778. (Contributed by BJ, 2-May-2021.)
⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Theorem	difex2 7780	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 7781	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 7782*	Expression for double union that moves union into a class abstraction. (Contributed by FL, 28-May-2007.)
⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}
Theorem	uniexr 7783	Converse of the Axiom of Union. Note that it does not require ax-un 7755. (Contributed by NM, 11-Nov-2003.)
⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
Theorem	uniexb 7784	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 7785	Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5365. (Contributed by NM, 11-Nov-2003.)
⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)
Theorem	pwexb 7786	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 7787	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 7788	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 7789	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	pwuncl 7790	Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)
⊢ ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)
Theorem	iunpw 7791*	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 7792	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 7793	A well-founded class 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 7794*	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)))
2.4.2  Ordinals (continued)
Theorem	epweon 7795	The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7755, see epweonALT 7796. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7755. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ E We On
Theorem	epweonALT 7796	Alternate proof of epweon 7795, shorter but requiring ax-un 7755. (Contributed by NM, 1-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ E We On
Theorem	ordon 7797	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 7798	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 7797), 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 7799	The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
Theorem	ssonuni 7800	The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 1-Nov-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))