Definition df-en 7027

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 7034. (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 7024	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1401	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1401	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1401	. . . . 5 class f
8	3, 5, 7	wf1o 5379	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1545	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4192	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1402	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is used by:  relen  7030  breng  7033  bren  7034  enssdom  7052