P. 65 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rdgruledefgg 6401*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
Theorem	rdgruledefg 6402*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ 𝐹 Fn V    ⇒   ⊢ (𝐴 ∈ 𝑉 → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
Theorem	rdgexggg 6403	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
Theorem	rdgexgg 6404	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ 𝐹 Fn V    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
Theorem	rdgifnon 6405	The recursive definition generator is a function on ordinal numbers. The 𝐹 Fn V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6412; in cases like df-oadd 6446 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
Theorem	rdgifnon2 6406*	The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
Theorem	rdgivallem 6407*	Value of the recursive definition generator. Lemma for rdgival 6408 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
Theorem	rdgival 6408*	Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
Theorem	rdgss 6409	Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))
Theorem	rdgisuc1 6410*	One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function 𝐹 other than 𝐹 Fn V. Given that, the resulting expression encompasses both the expected successor term (𝐹‘(rec(𝐹, 𝐴)‘𝐵)) but also terms that correspond to the initial value 𝐴 and to limit ordinals ∪ 𝑥 ∈ 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)). If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6411. (Contributed by Jim Kingdon, 9-Jun-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
Theorem	rdgisucinc 6411*	Value of the recursive definition generator at a successor. This can be thought of as a generalization of oasuc 6490 and omsuc 6498. (Contributed by Jim Kingdon, 29-Aug-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Theorem	rdgon 6412*	Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
Theorem	rdg0 6413	The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴
Theorem	rdg0g 6414	The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
⊢ (𝐴 ∈ 𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
Theorem	rdgexg 6415	The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 Fn V    ⇒   ⊢ (𝐵 ∈ 𝑉 → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
2.6.22  Finite recursion
Syntax	cfrec 6416	Extend class notation with the finite recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class frec(𝐹, 𝐼)
Definition	df-frec 6417*	Define a recursive definition generator on ω (the class of finite ordinals) with characteristic function 𝐹 and initial value 𝐼. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our frec operation (especially when df-recs 6331 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6423 and frecsuc 6433. Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4621. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6434, this definition and df-irdg 6396 restricted to ω produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.)
⊢ frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω)
Theorem	freceq1 6418	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))
Theorem	freceq2 6419	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐴 = 𝐵 → frec(𝐹, 𝐴) = frec(𝐹, 𝐵))
Theorem	frecex 6420	Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ frec(𝐹, 𝐴) ∈ V
Theorem	frecfun 6421	Finite recursion produces a function. See also frecfnom 6427 which also states that the domain of that function is ω but which puts conditions on 𝐴 and 𝐹. (Contributed by Jim Kingdon, 13-Feb-2022.)
⊢ Fun frec(𝐹, 𝐴)
Theorem	nffrec 6422	Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥frec(𝐹, 𝐴)
Theorem	frec0g 6423	The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
Theorem	frecabex 6424*	The class abstraction from df-frec 6417 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
Theorem	frecabcl 6425*	The class abstraction from df-frec 6417 exists. Unlike frecabex 6424 the function 𝐹 only needs to be defined on 𝑆, not all sets. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 21-Mar-2022.)
⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝐺:𝑁⟶𝑆)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝐺 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝐺‘𝑚))) ∨ (dom 𝐺 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ 𝑆)
Theorem	frectfr 6426*	Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and 𝐴 ∈ 𝑉 on frec(𝐹, 𝐴), we want to be able to apply tfri1d 6361 or tfri2d 6362, and this lemma lets us satisfy hypotheses of those theorems. (Contributed by Jim Kingdon, 15-Aug-2019.)
⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})    ⇒   ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V))
Theorem	frecfnom 6427*	The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)
Theorem	freccllem 6428*	Lemma for freccl 6429. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    &   ⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))    ⇒   ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Theorem	freccl 6429*	Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    ⇒   ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Theorem	frecfcllem 6430*	Lemma for frecfcl 6431. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)
⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))    ⇒   ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Theorem	frecfcl 6431*	Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.)
⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Theorem	frecsuclem 6432*	Lemma for frecsuc 6433. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.)
⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})    ⇒   ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Theorem	frecsuc 6433*	The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Theorem	frecrdg 6434*	Transfinite recursion restricted to omega. Given a suitable characteristic function, df-frec 6417 produces the same results as df-irdg 6396 restricted to ω. Presumably the theorem would also hold if 𝐹 Fn V were changed to ∀𝑧(𝐹‘𝑧) ∈ V. (Contributed by Jim Kingdon, 29-Aug-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))
2.6.23  Ordinal arithmetic
Syntax	c1o 6435	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 6436	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	c3o 6437	Extend the definition of a class to include the ordinal number 3.
class 3o
Syntax	c4o 6438	Extend the definition of a class to include the ordinal number 4.
class 4o
Syntax	coa 6439	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 6440	Extend the definition of a class to include the ordinal multiplication operation.
class ·o
Syntax	coei 6441	Extend the definition of a class to include the ordinal exponentiation operation.
class ↑o
Definition	df-1o 6442	Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
⊢ 1o = suc ∅
Definition	df-2o 6443	Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
⊢ 2o = suc 1o
Definition	df-3o 6444	Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 3o = suc 2o
Definition	df-4o 6445	Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 4o = suc 3o
Definition	df-oadd 6446*	Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
Definition	df-omul 6447*	Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
Definition	df-oexpi 6448*	Define the ordinal exponentiation operation. This definition is similar to a conventional definition of exponentiation except that it defines ∅ ↑o 𝐴 to be 1o for all 𝐴 ∈ On, in order to avoid having different cases for whether the base is ∅ or not. We do not yet have an extensive development of ordinal exponentiation. For background on ordinal exponentiation without excluded middle, see Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu (2025), "Ordinal Exponentiation in Homotopy Type Theory", arXiv:2501.14542 , https://arxiv.org/abs/2501.14542 which is formalized in the TypeTopology proof library at https://ordinal-exponentiation-hott.github.io/. (Contributed by Mario Carneiro, 4-Jul-2019.)
⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
Theorem	1on 6449	Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
⊢ 1o ∈ On
Theorem	1oex 6450	Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
⊢ 1o ∈ V
Theorem	2on 6451	Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ 2o ∈ On
Theorem	2on0 6452	Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
⊢ 2o ≠ ∅
Theorem	3on 6453	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 3o ∈ On
Theorem	4on 6454	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 4o ∈ On
Theorem	df1o2 6455	Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
⊢ 1o = {∅}
Theorem	df2o3 6456	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 2o = {∅, 1o}
Theorem	df2o2 6457	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
⊢ 2o = {∅, {∅}}
Theorem	1n0 6458	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
⊢ 1o ≠ ∅
Theorem	xp01disj 6459	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
Theorem	xp01disjl 6460	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
Theorem	ordgt0ge1 6461	Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
Theorem	ordge1n0im 6462	An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
⊢ (Ord 𝐴 → (1o ⊆ 𝐴 → 𝐴 ≠ ∅))
Theorem	el1o 6463	Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
Theorem	dif1o 6464	Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
Theorem	2oconcl 6465	Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)
Theorem	0lt1o 6466	Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ ∅ ∈ 1o
Theorem	0lt2o 6467	Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ ∅ ∈ 2o
Theorem	1lt2o 6468	Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ 1o ∈ 2o
Theorem	el2oss1o 6469	Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 15222. (Contributed by Jim Kingdon, 8-Aug-2022.)
⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)
Theorem	oafnex 6470	The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
⊢ (𝑥 ∈ V ↦ suc 𝑥) Fn V
Theorem	sucinc 6471*	Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)    ⇒   ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥)
Theorem	sucinc2 6472*	Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))
Theorem	fnoa 6473	Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
⊢ +o Fn (On × On)
Theorem	oaexg 6474	Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +o 𝐵) ∈ V)
Theorem	omfnex 6475*	The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V)
Theorem	fnom 6476	Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
⊢ ·o Fn (On × On)
Theorem	omexg 6477	Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ·o 𝐵) ∈ V)
Theorem	fnoei 6478	Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
⊢ ↑o Fn (On × On)
Theorem	oeiexg 6479	Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↑o 𝐵) ∈ V)
Theorem	oav 6480*	Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))
Theorem	omv 6481*	Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
Theorem	oeiv 6482*	Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
Theorem	oa0 6483	Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
Theorem	om0 6484	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
Theorem	oei0 6485	Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
Theorem	oacl 6486	Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
Theorem	omcl 6487	Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
Theorem	oeicl 6488	Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
Theorem	oav2 6489*	Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 suc (𝐴 +o 𝑥)))
Theorem	oasuc 6490	Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
Theorem	omv2 6491*	Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = ∪ 𝑥 ∈ 𝐵 ((𝐴 ·o 𝑥) +o 𝐴))
Theorem	onasuc 6492	Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵))
Theorem	oa1suc 6493	Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
Theorem	o1p1e2 6494	1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
⊢ (1o +o 1o) = 2o
Theorem	oawordi 6495	Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
Theorem	oawordriexmid 6496*	A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6495. (Contributed by Jim Kingdon, 15-May-2022.)
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 ⊆ 𝑏 → (𝑎 +o 𝑐) ⊆ (𝑏 +o 𝑐)))    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	oaword1 6497	An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
Theorem	omsuc 6498	Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
Theorem	onmsuc 6499	Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
2.6.24  Natural number arithmetic
Theorem	nna0 6500	Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)