Definition df-en 6827

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 6834. (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 6824	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1371	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1371	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1371	. . . . 5 class f
8	3, 5, 7	wf1o 5269	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1514	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4103	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1372	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  6830  breng  6833  bren  6834  enssdom  6852