Definition df-en 6800

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 6806. (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 6797	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1363	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1363	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1363	. . . . 5 class f
8	3, 5, 7	wf1o 5257	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1506	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4093	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1364	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  6803  bren  6806  enssdom  6821