Definition df-en 7956

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 7964. (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 7952	. 2 class ≈
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1482	. . . . 5 class 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1482	. . . . 5 class 𝑦
6		vf	. . . . . 6 setvar 𝑓
7	6	cv 1482	. . . . 5 class 𝑓
8	3, 5, 7	wf1o 5887	. . . 4 wff 𝑓:𝑥–1-1-onto→𝑦
9	8, 6	wex 1704	. . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦
10	9, 2, 4	copab 4712	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
11	1, 10	wceq 1483	1 wff ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}

This definition is referenced by:  relen  7960  bren  7964  enssdom  7980