Definition df-en 7069

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 7076. (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 7065	. 2 class ~~
2		vx	. . . . . 6 set x
3	2	cv 1648	. . . . 5 class x
4		vy	. . . . . 6 set y
5	4	cv 1648	. . . . 5 class y
6		vf	. . . . . 6 set f
7	6	cv 1648	. . . . 5 class f
8	3, 5, 7	wf1o 5412	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1547	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4225	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1649	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  7073  bren  7076  enssdom  7091