Definition df-en 7518

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 7526. (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 7514	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1436	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1436	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1436	. . . . 5 class f
8	3, 5, 7	wf1o 5536	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1657	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4417	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1437	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  7522  bren  7526  enssdom  7541