Definition df-k3

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)

	Ref	Expression
Assertion	df-k3	|- _K3 = { <. z , n >. | ( n e. 9o /\ E. x e. On E. y e. On z = ( _J ` <. x , y , n >. ) ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ck3	|- _K3
1		vz	|- z
2		vn	|- n
3	2	cv	|- n
4		c9o	|- 9o
5	3 4	wcel	|- n e. 9o
6		vx	|- x
7		con0	|- On
8		vy	|- y
9	1	cv	|- z
10		cj	|- _J
11	6	cv	|- x
12	8	cv	|- y
13	11 12 3	cotp	|- <. x , y , n >.
14	13 10	cfv	|- ( _J ` <. x , y , n >. )
15	9 14	wceq	|- z = ( _J ` <. x , y , n >. )
16	15 8 7	wrex	|- E. y e. On z = ( _J ` <. x , y , n >. )
17	16 6 7	wrex	|- E. x e. On E. y e. On z = ( _J ` <. x , y , n >. )
18	5 17	wa	|- ( n e. 9o /\ E. x e. On E. y e. On z = ( _J ` <. x , y , n >. ) )
19	18 1 2	copab	|- { <. z , n >. | ( n e. 9o /\ E. x e. On E. y e. On z = ( _J ` <. x , y , n >. ) ) }
20	0 19	wceq	|- _K3 = { <. z , n >. | ( n e. 9o /\ E. x e. On E. y e. On z = ( _J ` <. x , y , n >. ) ) }