Definition df-k3 35803

Description: Define a function that takes an ordinal and returns the third argument of the ordered triple ⟨𝑥, 𝑦, 𝑛⟩ such that the ordinal equals (𝐽‘⟨𝑥, 𝑦, 𝑛⟩). Based on the third case of Definition 15.7 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 2-Sep-2026.)

Assertion

Ref	Expression
df-k3	⊢ 𝐾3 = {⟨𝑧, 𝑛⟩ ∣ (𝑛 ∈ 9o ∧ ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

Distinct variable group:   𝑥,𝑛,𝑦,𝑧

Detailed syntax breakdown of Definition df-k3

Step	Hyp	Ref	Expression
1		ck3 35794	. 2 class 𝐾3
2		vn	. . . . . 6 setvar 𝑛
3	2	cv 1569	. . . . 5 class 𝑛
4		c9o 35690	. . . . 5 class 9o
5	3, 4	wcel 2145	. . . 4 wff 𝑛 ∈ 9o
6		vz	. . . . . . . 8 setvar 𝑧
7	6	cv 1569	. . . . . . 7 class 𝑧
8		vx	. . . . . . . . . 10 setvar 𝑥
9	8	cv 1569	. . . . . . . . 9 class 𝑥
10		vy	. . . . . . . . . 10 setvar 𝑦
11	10	cv 1569	. . . . . . . . 9 class 𝑦
12	9, 11, 3	cotp 4591	. . . . . . . 8 class ⟨𝑥, 𝑦, 𝑛⟩
13		cj 35791	. . . . . . . 8 class 𝐽
14	12, 13	cfv 6527	. . . . . . 7 class (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
15	7, 14	wceq 1570	. . . . . 6 wff 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
16		con0 6351	. . . . . 6 class On
17	15, 10, 16	wrex 3086	. . . . 5 wff ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
18	17, 8, 16	wrex 3086	. . . 4 wff ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
19	5, 18	wa 401	. . 3 wff (𝑛 ∈ 9o ∧ ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))
20	19, 6, 2	copab 5166	. 2 class {⟨𝑧, 𝑛⟩ ∣ (𝑛 ∈ 9o ∧ ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}
21	1, 20	wceq 1570	1 wff 𝐾3 = {⟨𝑧, 𝑛⟩ ∣ (𝑛 ∈ 9o ∧ ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

This definition is used by: (None)