Definition df-en 8958

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 8967. (Contributed by NM, 28-Mar-1998.)

Assertion

Ref	Expression
df-en	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃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 8954	. 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 6530	. . . 4 wff 𝑓:𝑥–1-1-onto→𝑦
9	8, 6	wex 1812	. . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦
10	9, 2, 4	copab 5167	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
11	1, 10	wceq 1570	1 wff ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}

This definition is used by:  relen  8962  breng  8966  enssdom  8987  enssdomOLD  8988