Definition df-oadd 6577

Description: Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)

Assertion

Ref	Expression
df-oadd	⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oadd

Step	Hyp	Ref	Expression
1		coa 6570	. 2 class +o
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4455	. . 3 class On
5	3	cv 1394	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2799	. . . . . 6 class V
8	6	cv 1394	. . . . . . 7 class 𝑧
9	8	csuc 4457	. . . . . 6 class suc 𝑧
10	6, 7, 9	cmpt 4145	. . . . 5 class (𝑧 ∈ V ↦ suc 𝑧)
11	2	cv 1394	. . . . 5 class 𝑥
12	10, 11	crdg 6526	. . . 4 class rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)
13	5, 12	cfv 5321	. . 3 class (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)
14	2, 3, 4, 4, 13	cmpo 6012	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
15	1, 14	wceq 1395	1 wff +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))

This definition is referenced by:  fnoa  6606  oaexg  6607  oav  6613