Definition df-kard 35818

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 35820 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35825. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 10020. (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 35817	. 2 class kard
2		vx	. . 3 setvar 𝑥
3		cvv 3451	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cen 8970	. . . . . 6 class ≈
8	5, 6, 7	wbr 5103	. . . . 5 wff 𝑦 ≈ 𝑥
9	8, 4	cab 2739	. . . 4 class {𝑦 ∣ 𝑦 ≈ 𝑥}
10	9	cscott 9928	. . 3 class Scott {𝑦 ∣ 𝑦 ≈ 𝑥}
11	2, 3, 10	cmpt 5186	. 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  35819  kardval  35820  kard0  35822