Definition df-kard 35959

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 35961 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35966. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 10057. (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 35958	. 2 class kard
2		vx	. . 3 setvar 𝑥
3		cvv 3454	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cen 9008	. . . . . 6 class ≈
8	5, 6, 7	wbr 5108	. . . . 5 wff 𝑦 ≈ 𝑥
9	8, 4	cab 2741	. . . 4 class {𝑦 ∣ 𝑦 ≈ 𝑥}
10	9	cscott 9965	. . 3 class Scott {𝑦 ∣ 𝑦 ≈ 𝑥}
11	2, 3, 10	cmpt 5192	. 2 class (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
12	1, 11	wceq 1570	1 wff kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})

This definition is used by:  kardfn  35960  kardval  35961  kard0  35963