Definition df-en 5427

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 5436.

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 5423	. 2 class ~~
2		vx	. . . . . 6 set x
3	2	cv 1297	. . . . 5 class x
4		vy	. . . . . 6 set y
5	4	cv 1297	. . . . 5 class y
6		vf	. . . . . 6 set f
7	6	cv 1297	. . . . 5 class f
8	3, 5, 7	wf1o 3997	. . . 4 wff f:x-1-1-onto->y
9	8, 6	wex 1326	. . 3 wff E.f f:x-1-1-onto->y
10	9, 2, 4	copab 3395	. 2 class {<.x, y>. | E.f f:x-1-1-onto->y}
11	1, 10	wceq 1298	1 wff ~~ = {<.x, y>. | E.f f:x-1-1-onto->y}

This definition is referenced by:  relen 5431  breng 5434  enssdom 5442

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