Definition df-en 8493

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 8501. (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 8489	. 2 class ≈
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1537	. . . . 5 class 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1537	. . . . 5 class 𝑦
6		vf	. . . . . 6 setvar 𝑓
7	6	cv 1537	. . . . 5 class 𝑓
8	3, 5, 7	wf1o 6323	. . . 4 wff 𝑓:𝑥–1-1-onto→𝑦
9	8, 6	wex 1781	. . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦
10	9, 2, 4	copab 5092	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
11	1, 10	wceq 1538	1 wff ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}

This definition is referenced by:  relen  8497  bren  8501  enssdom  8517