Definition df-en 6953

Description: Define the equinumerosity relation. Definition of [Enderton] p. 129. We define ~~ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 6960. (Contributed by NM, 28-Mar-1998.)

Assertion

Ref	Expression
df-en	|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

Distinct variable group:   x, y, f

Detailed syntax breakdown of Definition df-en

Step	Hyp	Ref	Expression
1		cen 6950	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1397	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1397	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1397	. . . . 5 class f
8	3, 5, 7	wf1o 5332	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1541	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4154	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1398	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  6956  breng  6959  bren  6960  enssdom  6978