Definition df-en 6797

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 6803. (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 6794	. 2 class ≈
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1363	. . . . 5 class 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1363	. . . . 5 class 𝑦
6		vf	. . . . . 6 setvar 𝑓
7	6	cv 1363	. . . . 5 class 𝑓
8	3, 5, 7	wf1o 5254	. . . 4 wff 𝑓:𝑥–1-1-onto→𝑦
9	8, 6	wex 1503	. . 3 wff ∃𝑓 𝑓:𝑥–1-1-onto→𝑦
10	9, 2, 4	copab 4090	. 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
11	1, 10	wceq 1364	1 wff ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}

This definition is referenced by:  relen  6800  bren  6803  enssdom  6818