Definition df-kard 35679

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 35681 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35686. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 9947. (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 35678	. 2 class kard
2		vx	. . 3 setvar 𝑥
3		cvv 3450	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cen 8952	. . . . . 6 class ≈
8	5, 6, 7	wbr 5103	. . . . 5 wff 𝑦 ≈ 𝑥
9	8, 4	cab 2738	. . . 4 class {𝑦 ∣ 𝑦 ≈ 𝑥}
10	9	cscott 9870	. . 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  35680  kardval  35681  kard0  35683