Definition df-card 9697

Description: Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 10302 for its value and cardval2 9749 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 10307. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)

Assertion

Ref	Expression
df-card	⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-card

Step	Hyp	Ref	Expression
1		ccrd 9693	. 2 class card
2		vx	. . 3 setvar 𝑥
3		cvv 3432	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1538	. . . . . 6 class 𝑦
6	2	cv 1538	. . . . . 6 class 𝑥
7		cen 8730	. . . . . 6 class ≈
8	5, 6, 7	wbr 5074	. . . . 5 wff 𝑦 ≈ 𝑥
9		con0 6266	. . . . 5 class On
10	8, 4, 9	crab 3068	. . . 4 class {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}
11	10	cint 4879	. . 3 class ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}
12	2, 3, 11	cmpt 5157	. 2 class (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
13	1, 12	wceq 1539	1 wff card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})

This definition is referenced by:  cardf2  9701  cardval3  9710  iscard4  41140  harval3  41145