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Definition df-oexpi 6416
Description: 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.)

Assertion
Ref Expression
df-oexpi o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oexpi
StepHypRef Expression
1 coei 6409 . 2 class o
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 4359 . . 3 class On
53cv 1352 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 2737 . . . . . 6 class V
86cv 1352 . . . . . . 7 class 𝑧
92cv 1352 . . . . . . 7 class 𝑥
10 comu 6408 . . . . . . 7 class ·o
118, 9, 10co 5868 . . . . . 6 class (𝑧 ·o 𝑥)
126, 7, 11cmpt 4061 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
13 c1o 6403 . . . . 5 class 1o
1412, 13crdg 6363 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
155, 14cfv 5211 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
162, 3, 4, 4, 15cmpo 5870 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
171, 16wceq 1353 1 wff o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
Colors of variables: wff set class
This definition is referenced by:  fnoei  6446  oeiexg  6447  oeiv  6450
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