Definition df-oadd 3106

Description: Define the ordinal addition operation.

Assertion

Ref	Expression
df-oadd	⊢ +o = {⟨⟨x, y⟩, z⟩∣((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = suc w}, x) ‘y))}

Distinct variable group(s):   x,y,z,w,v

Detailed syntax breakdown of Definition df-oadd

Step	Hyp	Ref	Expression
1		coa 3101	. 2 class +o
2		vx	. . . . . . 7 set x
3	2	cv 1089	. . . . . 6 class x
4		con0 2199	. . . . . 6 class On
5	3, 4	wcel 1092	. . . . 5 wff x ∈ On
6		vy	. . . . . . 7 set y
7	6	cv 1089	. . . . . 6 class y
8	7, 4	wcel 1092	. . . . 5 wff y ∈ On
9	5, 8	wa 196	. . . 4 wff (x ∈ On ∧ y ∈ On)
10		vz	. . . . . 6 set z
11	10	cv 1089	. . . . 5 class z
12		vv	. . . . . . . . . 10 set v
13	12	cv 1089	. . . . . . . . 9 class v
14		vw	. . . . . . . . . . 11 set w
15	14	cv 1089	. . . . . . . . . 10 class w
16	15	csuc 2201	. . . . . . . . 9 class suc w
17	13, 16	wceq 1091	. . . . . . . 8 wff v = suc w
18	17, 14, 12	copab 2055	. . . . . . 7 class {⟨w, v⟩∣v = suc w}
19	18, 3	crdg 2969	. . . . . 6 class rec({⟨w, v⟩∣v = suc w}, x)
20	7, 19	cfv 2422	. . . . 5 class (rec({⟨w, v⟩∣v = suc w}, x) ‘y)
21	11, 20	wceq 1091	. . . 4 wff z = (rec({⟨w, v⟩∣v = suc w}, x) ‘y)
22	9, 21	wa 196	. . 3 wff ((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = suc w}, x) ‘y))
23	22, 2, 6, 10	copab2 3002	. 2 class {⟨⟨x, y⟩, z⟩∣((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = suc w}, x) ‘y))}
24	1, 23	wceq 1091	1 wff +o = {⟨⟨x, y⟩, z⟩∣((x ∈ On ∧ y ∈ On) ∧ z = (rec({⟨w, v⟩∣v = suc w}, x) ‘y))}

This definition is referenced by:  fnoa 3117  oav 3119

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