Definition df-en 7013

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 7020. (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 7010	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1401	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1401	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1401	. . . . 5 class f
8	3, 5, 7	wf1o 5371	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1545	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4186	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1402	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  7016  breng  7019  bren  7020  enssdom  7038