Definition df-kard 35889

Description: Define the alternative cardinal number function. Under this definition, the cardinal number of a set is the set of all sets equinumerous to it and having the least possible rank. Definition of [Enderton] p. 222. See kardval 35891 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35896. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 10030. (Contributed by BTernaryTau, 2-Jul-2026.)

Assertion

Ref	Expression
df-kard	⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-kard

Step	Hyp	Ref	Expression
1		ckard 35888	. 2 class kard
2		vx	. . 3 setvar 𝑥
3		cvv 3453	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1568	. . . . . 6 class 𝑦
6	2	cv 1568	. . . . . 6 class 𝑥
7		cen 8981	. . . . . 6 class ≈
8	5, 6, 7	wbr 5105	. . . . 5 wff 𝑦 ≈ 𝑥
9	8, 4	cab 2741	. . . 4 class {𝑦 ∣ 𝑦 ≈ 𝑥}
10	9	cscott 9938	. . 3 class Scott {𝑦 ∣ 𝑦 ≈ 𝑥}
11	2, 3, 10	cmpt 5188	. 2 class (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
12	1, 11	wceq 1569	1 wff kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})

This definition is used by:  kardfn  35890  kardval  35891  kard0  35893