Definition df-en 9002

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 9011. (Contributed by NM, 28-Mar-1998.)

Assertion

Ref	Expression
df-en	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}

Distinct variable group:   𝑥,𝑦,𝑓

Detailed syntax breakdown of Definition df-en

Step	Hyp	Ref	Expression
1		cen 8998	. 2 class ≈
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1569	. . . . 5 class 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1569	. . . . 5 class 𝑦
6		vf	. . . . . 6 setvar 𝑓
7	6	cv 1569	. . . . 5 class 𝑓
8	3, 5, 7	wf1o 6557	. . . 4 wff 𝑓:𝑥⟶1-1onto𝑦
9	8, 6	wex 1812	. . 3 wff ∃𝑓 𝑓:𝑥⟶1-1onto𝑦
10	9, 2, 4	copab 5170	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}
11	1, 10	wceq 1570	1 wff ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}

This definition is used by:  relen  9006  breng  9010  enssdom  9031  enssdomOLD  9032