Definition df-en 7514

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 7522. (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 7510	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1378	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1378	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1378	. . . . 5 class f
8	3, 5, 7	wf1o 5585	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1596	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4504	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1379	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  7518  bren  7522  enssdom  7537