Definition df-kard 35563

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 35565 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35570. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 9921. (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 35562	. 2 class kard
2		vx	. . 3 setvar 𝑥
3		cvv 3455	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cen 8936	. . . . . 6 class ≈
8	5, 6, 7	wbr 5109	. . . . 5 wff 𝑦 ≈ 𝑥
9	8, 4	cab 2741	. . . 4 class {𝑦 ∣ 𝑦 ≈ 𝑥}
10	9	cscott 9853	. . 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 referenced by:  kardfn  35564  kardval  35565  kard0  35567