Definition df-en 7516

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 7524. (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 7512	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1380	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1380	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1380	. . . . 5 class f
8	3, 5, 7	wf1o 5574	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1597	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4491	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1381	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  7520  bren  7524  enssdom  7539