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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) | ||
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)) | ||
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)) | ||
Theorem | nna0 6500 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) |
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