Definition df-en 7510

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 7518. (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 7506	. 2 class ~~
2		vx	. . . . . 6 setvar x
3	2	cv 1397	. . . . 5 class x
4		vy	. . . . . 6 setvar y
5	4	cv 1397	. . . . 5 class y
6		vf	. . . . . 6 setvar f
7	6	cv 1397	. . . . 5 class f
8	3, 5, 7	wf1o 5569	. . . 4 wff f : x -1-1-onto-> y
9	8, 6	wex 1617	. . 3 wff E. f f : x -1-1-onto-> y
10	9, 2, 4	copab 4496	. 2 class { <. x , y >. | E. f f : x -1-1-onto-> y }
11	1, 10	wceq 1398	1 wff ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }

This definition is referenced by:  relen  7514  bren  7518  enssdom  7533