Definition df-card 7157

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. 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 7156	. 2 class card
2		vx	. . 3 setvar 𝑥
3		cvv 2730	. . 3 class V
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1347	. . . . . 6 class 𝑦
6	2	cv 1347	. . . . . 6 class 𝑥
7		cen 6716	. . . . . 6 class ≈
8	5, 6, 7	wbr 3989	. . . . 5 wff 𝑦 ≈ 𝑥
9		con0 4348	. . . . 5 class On
10	8, 4, 9	crab 2452	. . . 4 class {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}
11	10	cint 3831	. . 3 class ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}
12	2, 3, 11	cmpt 4050	. 2 class (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
13	1, 12	wceq 1348	1 wff card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})

This definition is referenced by:  cardcl  7158  isnumi  7159  cardval3ex  7162