Definition df-k3 35927

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 35918	. 2 class 𝐾3
2		vn	. . . . . 6 setvar 𝑛
3	2	cv 1568	. . . . 5 class 𝑛
4		c9o 35814	. . . . 5 class 9o
5	3, 4	wcel 2144	. . . 4 wff 𝑛 ∈ 9o
6		vz	. . . . . . . 8 setvar 𝑧
7	6	cv 1568	. . . . . . 7 class 𝑧
8		vx	. . . . . . . . . 10 setvar 𝑥
9	8	cv 1568	. . . . . . . . 9 class 𝑥
10		vy	. . . . . . . . . 10 setvar 𝑦
11	10	cv 1568	. . . . . . . . 9 class 𝑦
12	9, 11, 3	cotp 4591	. . . . . . . 8 class ⟨𝑥, 𝑦, 𝑛⟩
13		cj 35915	. . . . . . . 8 class 𝐽
14	12, 13	cfv 6538	. . . . . . 7 class (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
15	7, 14	wceq 1569	. . . . . 6 wff 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩)
16		con0 6362	. . . . . 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 1569	1 wff 𝐾3 = {⟨𝑧, 𝑛⟩ ∣ (𝑛 ∈ 9o ∧ ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}

This definition is used by: (None)