Definition df-card 7782

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 8377 for its value, cardval2 7834 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 8382. 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 e. _V |-> |^| { y e. On | y ~~ x } )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-card

Step	Hyp	Ref	Expression
1		ccrd 7778	. 2 class card
2		vx	. . 3 set x
3		cvv 2916	. . 3 class _V
4		vy	. . . . . . 7 set y
5	4	cv 1648	. . . . . 6 class y
6	2	cv 1648	. . . . . 6 class x
7		cen 7065	. . . . . 6 class ~~
8	5, 6, 7	wbr 4172	. . . . 5 wff y ~~ x
9		con0 4541	. . . . 5 class On
10	8, 4, 9	crab 2670	. . . 4 class { y e. On | y ~~ x }
11	10	cint 4010	. . 3 class |^| { y e. On | y ~~ x }
12	2, 3, 11	cmpt 4226	. 2 class ( x e. _V |-> |^| { y e. On | y ~~ x } )
13	1, 12	wceq 1649	1 wff card = ( x e. _V |-> |^| { y e. On | y ~~ x } )

This definition is referenced by:  cardf2  7786  cardval3  7795